New to airguns, need suggestions.

I'm rather new at this as well and I am certain that other more experienced people will respond, but the short answer is that any of the above mentioned will do the job. A lot depends on how much money you are willing to invest. The least would be springers, or gas piston break barrels. Next are CO2 which requires you to purchase gas cylinders in addition to pellets. PCP's are the top of the line and require an air source for filling, usually a high pressure tank for refilling. 

If money is not an issue a PCP with a carbon bottle will get you roughly 80 shots depending on caliper. Trust me a good gun with a moderator will be really quiet from your previous springer choice. PCPs are a pleasure to shoot.

Just how many squirrels do you see on your walk? I do have a Benjamin Marauder pistol, and I can get about 30 shots 1 full tank. Do you see more than 30 squirrels in your walks?

"YmF8M86"Thanks for the suggestions. I am not price sensitive, anything is fine. However, I will be walking through the woods, and it seems strange to have to take a high-pressure tank with me in the woods to hunt squirrels. That certainly did not exist back in the 70s. Why would someone want to do that?

Rather than just mail order them and then find out that what I ordered is horribly inappropriate for what I am trying to do, I am just hoping to get some guidance. My second attempt has been to order a Benjamin Marauder Woods Walker PCP .22 from Amazon. Hopefully that will be more appropriate for a day of squirrel hunting, if I can figure out how to take the High Pressure Hand Pump with me into the field.

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That should give you around 30 shots before needing a refill. You could get a small carbon fiber bottle, weighing under 5 lbs, that you can carry in a backpack rather than using a hand pump. Much quicker and more convenient for refilling.
http://www.airgunsofarizona.com/air-cylinders/omega-12-cu.-ft.-4500-psi-portable-air-tank/

If you only want something to plink with ot shoot some squirrels and rabbits then a .177 or a .22 is a good choice, now with that said there are PCP that will get 60-100 shots on a single air fill so you need to only carry the Airgun in the woods, there are a few realy good starter packages out there that come with everything needed to get you out shooting, For example the Benjamin Marauder in .177 or .22 is probably the most bought Airgun out there, it’s very accurate and has a good reputation, you can get a package that has the Airgun,handpump, pellets and scope with rings for around 500 dollars but look around.This Airgun is easy to fill with a handpump and you’ll get more then enough shots for a walk thru the woods.
The next step would be to buy a airtank and fittings to fill your Airgun so you don’t need the handpump but when you step up to a airtank you need a source for air like a scuba shop or local fire department that has the means to fill your tank to 4500psi....It very critical that before you buy a airtank you 1st find a place that can fill to 4500psi, if you can’t find a place then you need to either buy a Shoebox air pump or a Regular PCP Compressor which can be anywhere from a cheap EBay 360.00 Compressor to a 1299.00 AV or up to 4-5k.....
So you see there are a lot of ways you can go when you get into this hobby or sport it just depends on how much your willing to spend or can afford...If you decide Airgunning isn’t for you, you can recoup most of your investment by selling everything and mark it up to a adventure in life.
Good luck on your journey into airguns......

"YmF8M86"What is a carbon bottle? How much does it weigh? Do you carry it with you in your pack?
What is a moderator?
Are there any advantages of a PCP over the other technologies? Are they more accurate? Quieter? Lighter?
Maybe I should just buy a cheap BB gun like I used to have in the 1970s, but I don't know if those could kill a squirrel or would be very accurate.

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Guns with a carbon bottle air reservoir that is installed on the gun. They can be also unscrewed and replaced with a spare if need be. A moderator is used to silence the gun. Most guns sold in the US have some kind of moderator sold as part of the gun package.

When I started in the hobby I had a Air Arms TX200 Spring gun rifle which is one of the best in it's class. I then got hooked on airguns shortly after that. When I decided to make the jump into PCP rifles I decided I never wanted to fill a gun with a hand pump. I then saved my money and bought an AA S510, and at the same time I purchased a scuba bottle and Shoebox air compressor. Big investment at first but that purchase was out of the way. If I would have started with less I would have blown money trying to eventually get there. Everyone with different budgets has their own way of getting to a place in the hobby. I love shooting these PCP airguns for their quietness and accuracy. I can do all this from 20 feet out my backdoor. Never could do that with a cartridge/powder burner gun.

*Edit While typing my last response. I see you said you bought a pistol and a pump. A pump should be good for a pistol since they won't take much air. The fill per shots will be cut down as well due to the small air reservoir. It will also be louder without the longer barrel that keeps airguns quieter.

"YmF8M86"Thanks lmold. For my second attempt, I have ordered a Benjamin Marauder Woods Walker PCP .22 pistol with a Benjamin High Pressure Hand Pump. I have no idea what to expect, since I have never used a PCP gun before. I don't know if it will be loud, heavy, accurate, or none of the above, compared to other choices. I don't know why PCP is considered top of the line if you have to buy a lot of equipment just to charge them. What are the advantages?

Do you think that the gun I have ordered will be able to kill 2-3 squirrels in a day of hunting while being reasonably quiet?

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I don't own one but I understand that they are very quiet. Like I said before expect to get around 30 good shots per fill.

I’m fairly new to this too, fairly late in life, and I don’t know as much as most guys here. I can give you some input though, from my point of view. 

The name of the game here is accuracy, everything else comes second. 
You want to walk in the woods? Lightweight will help. 
I have a bias against CO2, so, no comment. 
Spring powered break barrel or underlevers will be proportionately heavy as power increases. I would stick with quality brands. You don’t need a lot of power, just YouTube British airgun hunters. HW30 is a favorite with folk. Heavy helps with accuracy on the more powerful guns. 
PCPs are great, quiet, accurate, expensive, but the biggest problem is having a way to get air in them. 
Marauders are a favorite as well, but most people I know have to tinker a lot with them before they are performing as they want them too (I have never owned one) 
A good scope is a must. 
Consider what distance you are shooting at, a lot of guns will do, if 30 yards or less is all you shoot. 
You will need to research and test different pellet brands, weights, shapes to find the ONES your gun likes. 

Hope some of this helps. 

So I'm into my second year of the pellet gun world, same as you I asked the same questions, I wish I had bought the best stuff 1st and not waisted my money on the other guns I still own. So that said, do you have a place like a scuba shop or your local fire dept that can fill a air tank?, is so buy a tank, if not buy a compressor or you will be hand pumping, then buy a good pcp air rifle, there are many, I have a daystate regal .22, fx wildcat .25 and a taipan mutant .22, all are insanely accurate, the taipan was 1st suggested to me but I did not like the looks of it, that aside it needed nothing but air and lead to shoot hole on hole at 40 yards and better, same with the cat and regal. If you want to play more and like to tinker there are other pcps such as the marauder which I own, love and have had modded to work the way it should from the factory. I have expensive springers too, they just don't cut the mustard, once you go pcp there is no going back. The same goes for optics, I have a drawer full of cheap scopes, buy good optics, you don't get prime rib for hamburger prices, This is just my 2 cents worth, other people willl offer up other ideas. This is a great site for info, lots of good people, good luck with the decision process.

Let start with some basic powerplant explanation. The spring or gas piston guns (like your Diana 350) can be very accurate in the hands of an experienced shooter. They use a powerful spring or gas piston to compress air in the chamber and propel the pellet. It requires some practice to develop a consistent hold to get the best results. Their realistic effective range in the hands of an average shooter is out to 35-40 yards. Benefits are they're self-contained and simple to operate. The more mild powered versions are much easier to shoot as a rule then the super powerful models which are often scope killers due to the severe recoil. The dual forward and backward recoil of a spring or gas piston gun requires an "airgun rated" scope.
Something like an HW95/Beeman R9 or an Rws 34 will make a great walk around hunting rifle.
By the way, it is very unwise to fire a spring or gas piston gun without a pellet loaded. It will very like damage the internals of the gun. You mentioned a very loud report when you test fired the gun-this will happen if the gun is dry fired! It's also common for excess lube to ignite from pressure during the break-in period on a new gun, causing a louder report and a puff of smoke from the barrel. This is called dieseling and should go away after shooting 100 or so pellets.
Co2 guns are relatively simple to operate but require powerlets or a co2 tank of some kind as propellant. They are easy to shoot accurately but don't develop that much power due to the way Co2 expands. also, the point of impact will vary with temperature. They are effective on small game (with a well placed shot) out to 20-25 yards. There is no recoil, so any scope will work. All you need is your gun, some pellets and a few powerlets or a tank of Co2 to shoot.
Precharged pneumatic (PCP) guns used compressed air to propel the pellet. Most have an airtube or tank attached to gun which provides enough air for as few as 3 or as many as 100 shots. They are generally very accurate and easy to shoot accurately with effective ranges out to 50 yards and beyond depending on the type of gun and the shooter's experience. 
PCPs can be rather expensive and require a source of air such as a special hand pump or a scuba or SCBA tank.
A great starter PCP is the Benjamin Marauder. I'd recommend you steer clear of the really complicated PCPs until you get a some experience, just because a gun is expensive doesn't necessarily make it better or more accurate than a more basic model! Once you have a source for air, you will likely wind up with more than one gun since they're so much fun to shoot!
Whatever you choose, have fun and shoot safely!
John
PS- just saw you last post, the Marauder pistol is really fun gun, most folks seem to get best results with H&N Field target trophy pellets in the larger (5.54) head size. You can order different head sized H&Ns from Pyramyd Air. Some Marauder pistol owners may chime in with what works best for their guns.

Where I live squirrel hunting is mostly a 40 yard shot as the woods around here are pretty thick I very really take a longer shot for squirrels. I use a Edgun Leshiy which is a small foldable rifle that only weighs a few pounds. On a full tank I get 35 shots in 22 and if I switch to a 25 caliber barrel (only takes 2 minutes) the count drops to 23. The gun only weighs about 4pounds a little over 5 with scope. I also carry in my backpack a 90 cc tank that will refil as much as you can hunt in a day.
my first pcp rifle with a scope was Hatsan Gladius a bulpup design its a good shooting gun but it weighs 12+ pounds with scope that is too heavy for hunting unless your posting up all day. 
For under 75 yards the Leshiy is my pick you can hang it off your belt while walking to the hunting grounds. Check it out or see if someone around you has one you can look at.
check this video for Leshiy in action
https://youtu.be/dSCDT-X3Chk

Springers are hold sensitive and have significant recoil due to the spring's action when released. PCPs are kinda like a .22 long rifle, almost no kick if any, not hold sensitive at all, tend to be much lighter than their spring counterparts. They can be silenced effectively. The good ones will put 10 pellets or more inside of a dime or less at 30 to 50 yards. They usually are multi-shot guns that can use a magazine for quick follow up shots. They are available in Semi-automatic (not my cup of tea) 

I suggest you research the following guns: Daystate Regal XL, Kalibrgun Cricket, Taipan Veteran, Vulcan by Air Tech, Renegade or Pulsar by Daystate (the later is electronic) May want to check out the new Brocock Compato Target. Don't think you can go wrong with any of these. I have a Cricket, and the thing is a lazer. 

Currently I'm using the Sightron S-Tac very good scope. Look up Georgia Airguns (Charlie Frear) look for Ernest Rowe as well, he is in NJ.