Just my 2cents worth.
Most have made one very good point, go and see if it is something you want to spent time doing.
Borrow a gun (doesn't matter what caliber if you aren't worried about the pets) or us a shot gun as it sound you will be in close quarters.
If you want a, one gun does all, my personal favored is the 3006 (most everything has been taken with one, I wouldn't suggest using it on large bears though, even though they have been taken with the 3006) But it also has significant felt recoil.
The 243 is also a nice caliber for varmint shooting and small deer out to 200 maybe 250 with well placed shoots (shoot placement is everything no matter what caliber though) I use mine mostly on PD's.
If you want it strictly for a large varmint gun (here's were I'm going to get FLAMED) I would stick with a .223 or a 22-250 and the only reason is the large selection on factory ammo you can purchase and if you want to start reloading, there is an endless selection of bullets and powders to choose from.
If it is something you want to do, Purchase something that is rugged and you aren't worried about getting scratched up, But I wouldn't buy the cheapest I could find either.
Glass for the top don't scrimp here, you'll never be happy with a fussy cheap piece of glass on top (this will be a whole new post you will want to do.
Good luck and keep your powder dry
OH! almost forgot if you want it for PD's or sage rats that is a whole new ball game in its self