Yeppers, shooting indoors is cool. If you go .177 and a lower power rifle or pistol its definitely okay. Cheap and easy targets/pellet traps: I lean a 4ftx4ft piece of 3/4inch plywood up and then lean a 4ftx4ft piece of 2 inch thick ridged insulation over theat, then tack your paper targets onto that. the plywood is a good backstop to prevent the pellets from escaping, the insulation prevents bounce back/ricochets. i use all sorts of things for paper targets, from junk mail, old cereal boxes, old printer paper. You will find yourself shooting alot more if you arent counting $ for targets.
There are several classes of airguns:
Springers, you cock the gun the engage a spring or gas piston, when you pull the trigger the piston releases and rapidly compresses the air shooting the pellet out. Pros: lots of options for buying, no exterior air source needed, simple/easy maintenance, some of the best prices you will find. Cons: recoil and vibration (nothing like firearms, but compared to most other airgun options springers recoil most), if you want a springer that is going to be accurate well out past the 5/10 yard range and get good groupings pony up the cash and buy quality.
Co2: air guns that use co2 cartridges to operate. Pros, small, light, lots of options many are very affordable, some are full auto. Cons: you have to have a nice stash of co2 catridges and pellets to play, usually lower fps, mainly for target practice and plinking.
Multi pumps: a gun that has an onboard pump that you pump multiple times to fill the tank with varying air pressure. Pros, fully self contained like springers with much less to no recoil, you can pump once or up to 8 times (depends on the gun) to get the power you need, makes multipumps more versitile to go from indoor plinker to outdoor rodent hunter. Cons, you have to pump multiple times for one shot, not too many options for multi pumps that i know of out there. Daisy has the corner on them, but the Seneca Dragonfly is a nice multi pump.
Single Stroke: Like the multi pump, but you only pump it once. Pros, usually very accurate and reliable. Cons: lower fps (good for indoors though), usually more costly, only good for target shooting.
PCP: pre compressed pneumatics. these guns have an air chamber on them that you pressurize before hand and will shoot with many guns 20+ shots before refilling. Pros: very little recoil, multiple shots, can fill with a hand pump for complete independence. Cons, pcps tend to be more costly, and you need to buy extra equipment (hand pump, or air bottle that will need to be filled at a scuba shop, or a high pressure compressor) which adds to the start up cost.
On board pump Pcps: all the benefits of pcps, with an onboard hand pump for total independance. Cons: very few buying options only two that i know of: the FX Indy is excellent but way expensive. The other is the Nova Freedom, which is comparably priced to many mid range pcps.
So really you got to ask yourself what you want, what you can get, and what you will use. By the sounds of it a sringer or co2 will be the best go for you as they are the most cost effective, but with any airgun be ready to hunt for that right pellet, most pellet guns have a particular brand and weight of pellet it likes. North American Pellet Swap is a good facebook group to trade out pellets to land a variety of pellets without dropping lots on a bunch of ammo that your gun doesn't like.
My personal recommendations for particular guns are:
pcp: benjamin marauder pistol (woods walker) an excellent mid range/starter pcp, that has a good trigger, good accuracy, compact, lots of aftermarket parts so you can upgrade it as you fall deeper into the sport
Springer; i own a HW30S in .177 accurate and a great gun, almost everyone will agree with it, but you will be looking around $300. Krale airsports is base in europe and has some of the better prices on the Weihrauch springers, but Pyramyd air is competitive with their pricing on the HW30s in particular.
Single stroke: the Daisy 753S Match Grade Avanti for rifle, i own the Chiappa FAS 6004 (pistol .177) and love it, it's one of my favorites
Most of those are in the 300-400 range, there are plenty of break barrel springers out there for less. If you are going cheaper with springers i would recommend going for an under lever. the advantage to break barrels is they are easier to load, but you are moving the barrel everytime and that is why i bought the Weihrauch for a break barrel, pay for quality.
definitely shop online, the selection is too good to do anything else. I have been extremely happy shopping with Pyramyd Air. Krale has also been good to me, and TP3 Ranch by Thurmond Moore (look for link on the forms scrolling banner, blue circle with a bull sketch) has gotten me some good deals and a great guy to work with.
hope some of that helps