You're on the right path. You got the right general idea as to difference between springers and PCPs.
PCPs are generally more accurate, quieter, and have further range than springers. Springers have quirks such as a double recoil that can kill a scope and can be sensitive for accuracy purposes as to how you hold them. The primary advantage to a springer is that you don't have to fool with an air source. I generally steer people towards PCPs whenever their budget allows.
My first PCP was a .25 Benjamin Marauder and I never regretted the purchase. Its a hyper accurate and quiet gun out of the box. There's lots of ways you can tinker with it too should you decide to later. The only drawback I can see to starting with one is that its a hefty gun. There are options to make it lighter such as aftermarket aluminum air tubes, but that wouldn't be something to mess with for your first gun. There's been a couple of minor redesigns/upgrades to the Marauder model since it was introduced almost a decade ago. You'd want to get one of the newest ones. With coupons or deals you can expect to pay $500-$550 for one, leaving you room in your budget for an air source.
Over the past year I've also come to like the Airforce guns. You'd probably be interested in one of the SS models in either Condor or Talon builds (they come with a suppressor and are generally set up to be good pellet shooters, as opposed to other Airforce guns that are very good slug guns for larger game). They have a wider range of power adjustments and will be more versatile than a Marauder over a wider range of game. However they are single shot guns (meaning you have to manually reload each shot, not that you only get one shot on a fill of air, you'll get lots of shots on a single fill of air). The Marauders come with rotary magazines that can give you multiple shots with minimal movement. A Condor SS in .25 will be a couple of pounds lighter than a Marauder, will get far more shots on a fill, and can be set for almost twice the max power of a .25 Marauder.
Someone with more experience with the Hatsan guns can weigh in on those. I have no experience with them (see below post).
I also have one of the budget PCPs in .22 (a Gamo Urban) and have owned another (a Benjamin Discovery) but gave it away as a gift. Both are fine .22 airguns for their cost. But the Discovery requires an aftermarket suppressor. The Urban has one built on and is also magazine fed, while the Discovery is single shot.
I would recommend a .25 in whatever you get, so long as your primary purpose isn't shooting in the backyard where a .25 can pass thru most backstops. The .25 has a lot more oomph for hunting than a .177 or .22. The .25 is more impressive to people for their first hunting airguns. You'll be amazed how sometimes the .25 can kill better than a .22LR rimfire that has far more energy.
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