The Thompson Center Contender is what I have used primarily. I also use a S&W Mod 29 6" in .44 magnum. You can pick up a T/C frame in the area of $250 or so. Barrels can be had in a variety of cals for around $175-$225 depending, I have two barrels, one in 30-30 Winchester and one in .221 Fireball. You can pick up a frame and then get a barrel later... not that much cash put out at one time. Either use iron sights, which most barrels will have, or go with a scope... extended eye relief pistol scopes in 1x4 or 2x7 type will do great.
With the Contender, you can get a .35 Remington barrel or a .30-30 Winchester barrel and be quite deadly. You can step up to the Encore frame and get other cals such as .243 Winchester, 7mm-08 Rem, .260 Rem, etc.
I've taken a nice 6-pt with the 30-30 barrel using hand loaded 125gr Nosler BT's.
I'm thinking a used T/C in 30-30 Winchester with a 2x scope ought to get you going for a minimum investment.