A bunch of good advice so far. If you are in relatively good condition you can use a pump, but it is slow and the pump heats up, so every 500 PSI over 1000, you need to let the pump cool down. It'll also give you a break if you need it.
Depending on your use .22, .25, .30, are all good chamberings, and they get more expensive in that order. I tried the FX slugs and they shot patterns out of my rifle.
Nielsenspecialtyammo.com sells much better slugs and at a better price. JSB Match, and Exact, are great pellets, most others are hit or miss in my experience.
My best rifle will do 1/4MOA at 50 yards, is .22 and is as quiet as snapping your fingers. For simple target shooting indoors my HW 30 in .177 will do 1/2 MOA at 25 yards and less and it's a break barrel springer so no pump required. PCPs are nice because they don't require special scopes like springers and gas rams. I have friends that have China made compressors in the $300 to $500 range and they like them. I just try to avoid Chinese anything whenever possible.
As far as pest removal, PCPs are my only choice. I'm a good shot but for whatever reason I need more than I'd expect to get humane kills. My .20 PCP will pass through a squirrel out to 35, and my .22 out to 55 and I always make head shots. Others can probably do better but I know my limitations. I only use my .25s for target practice out past 50 yards, and they get MOA or better to 100 yards. I would expect them to take small game like raccoon and fox out that far but I don't get that many shots that for out in my dense woods.
I hope this helps. Like others, you can PM me for any questions.