I was in the same boat at the start of lock down and I raided the kid's piggy-banks to buy an SMK Artemis CP2, Cheap, runs on CO2, very quiet and you can swap out the barrels to have a pistol, a very short carbine (ideal ratting gun in sheds) or a rifle (don't make the long-barrelled pistol combination, it is over-powered). Great for plinking and short-range ratting but not that powerful, the rifle is about 8.5 ft lb.
I then sent off for a bunch of parts from T R Robb (
https://www.trrobb.com) such as hammer springs, transfer ports, spring guides and such which both gave me a good understanding of air-gun tuning without ruining an FX and the power is now close to 11ft lb, which is good enough for rabbits and pigeons out to 25m or so. Total cost for the gun and the bits was about £200.
As to calibre, there are so many arguments going and everyone has an opinion. Both .177 and .22 work well in sub-12 ft lb guns. I bought a .22 and now want a .177 too for those shots that you can't make with a .22. I'd probably suggest that if you are shooting in your garden and only want to target rats the .22 is probably best, you get quite a thump to the target but the pellets won't over-penetrate and take out the neighbours cat through the larch-lap.
Any good?