There was a fad here of shooting pigs with break barrel air guns, i know of exactly ONE instance where it was a success...dont do it.
If your set on building an air rifle hunting battery youll want atleast 2 guns. A "light" or "medium" for small game and plinking, and a "heavy" for small medium game, Im not a fan of the idea of taking larger game with an over the counter airgun even tho its done regularly.
Any .22 caliber that delivers 650+fps with 13grn pellets should be capable of taking squirrels at reasonable ranges, but If your looking at shooting stuff larger like big rabbits, I would want a gun that delivered at least 25gr pellets at 650+ so heavy .22 or .25+.
If your going for stuff larger than that your squarely into the medium bore PCP airgun territory.
I would probably recommend a Benjamin Marauder for a medium cost rifle (you need a pump and or airtank set up as well), and a Trail XP for a low cost option....most likely id get either in .25
https://www.pyramydair.com/a/Air_guns/588/ru_1/calibers_0_45/calibers_0_357/calibers_0_5 those are a list of guns, just a general list, that should be capable of taking small medium game with good shot placement at short range. I didnt look thru it completely, but you want to have at least 300fpe at the muzzle and be shooting a heavy solid. Also understand that as with archery, your not likely to get a DRT reaction from a critter hit by anything but the most powerfull of airguns, and even then its pretty rare.
I would probably get an airforce TexanSS
https://www.pyramydair.com/s/m/AirForce_TexanSS_Big_Bore_Air_Rifle/4317
The cheapest option i think i would choose would be a 909 in .45.
To go along with the high power airgun your going to at least need a pump, but more likely a set of refillable tanks. these would be useable on a lighter pcp as well.
Airguns are fun and cheap to get into, but when your looking at achieving the kind of power and penetration you need to securely take big game your cost increase exponentially.
I did the pcp thing for a while, its alot cheaper now, and were i getting into hunting with airguns seriously id would probably start with pcps.