No semiauto rifles or shotguns.
Shotguns you are lookigat pumps. Something inexpensive for home would be a Mosberg, Itaca, Remington or Winchester 18in to 20in in 12 or 20 even 410 GA. Lots of used at decent prices. I think a new Mosberg Maverick is still under $350.00.
Rifles, bolt action, price depends on caliber
Lever action or pump. Once again depends on calibers. Any brand used look for bargains. With an FID your choices are very limited to no semiautos.
Last I looked, Maverick 88 with the field/security barrel option MSRP is $293, and basic all-purpose ones MSRP for $262. It's a really great value, and those aforementioned packages can be had for $250 and $220 on sale respectively. You can also turn a Maverick 88 into a 590 clone or a tacticool set up without running afoul of anything for an extra $120, replete with pistol grip, adjustable A2 stock and a pump grip with rails for mounting a foregrip.
Some other suggestions I'd put up.
Henry Lever Golden Boy .22LR can be had new for under $500. Henry's are great guns, and you can drop down to the $290-$350 range and get a Rossi or Chiappa, which are perfectly usable but not anything special. PSA has a labor day sale for Henry's basic H100L (basically a Golden Boy without the gold and gorgeous walnut stock) and honestly is a great plinker. While this sale is going on, Rossi and Chiappa aren't worth considering unless very cheap and used. Golden Boy's with their walnut stocks are worth a second look despite being an extra $150. You get what you pay for.
There's also the Heritage Rough Rider Cowboy Carbine Revolver for $360. This is less practical, as anything past the 8" version actually loses muzzle velocity, but it wins points back on the cool factor alone.
If you want something with actual nut in it, PSA also has a G-force .357 Magnum lever gun for $500 from PSA, also on sale for Labor day with a 20" barrel and ammo made for longer rifle-length barrels, 2000 fps is easily attainable, which means you can bang steel out to 300 yards. I stress the "can" and not so much "will be able to". .357 while great, really needs hand-loads for repeatable accuracy. G-force is newer to the market, and I personally have only shot a few of their semi-auto shotguns and other imported oddballs, and the consensus is that they usually work just fine, and if they don't, G-Force will make them work for you.
Past $500, you start getting into .308 and 6.5 CM and the world kind of opens up, where your wallet is the only determining factor in limiting what you can get. Be sure to budget accessories into whatever direction you go in, otherwise you'll be wondering why you put a $80 airsoft sight onto your $500 gun.