Shotgun get a model 1100 Remington 12 gauge. They are plentiful used and if something goes wrong, which is highly unlikely, any gun shop will have or can get the parts needed to fix it.
Rifle if you want to save money get a Ruger American in 30-06 or 308 if you don't mind a little kick or 243 or 6.5 Creedmore if you want mild recoil. You can spend a lot more for a rifle but the Ruger American, on sale right now for $279 at Cabelas, is as accurate as any of them. As someone said get a Leupold scope. 4-12 would be good IMO.
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