Lever action - Winchester 92 (or the 1892 – but everyone calls them 92’) – chamber in either 357 magnum, 44 magnum or .454 casull (all you need is a cowboy hat and you will look like Clint Eastwood or John Wayne)
Ruger Mini-14 – chamber .223 – decent price and last forever (almost as good as the AK)
Springfield M1 Grand – chamber in .308 – freaking rifle is awesome and good, solid feel
Kel-Tec SU16C – chamber .223 – cheap, cheap for around than $500 – feel cheap (because of plastic) but good starter semi.
You can also get the Bushmaster or DPMS for .223 semi-auto rifle, with the banana clip, you can get 30 rounds
Unfortunately the Baer, Wilson, FNH or Browning will be over your $1k price range.
By the way, stick with .233 or 308 (most common) – if you start going into other, such as 30-30, 30-60, 7.62*45 – the cost of ammo will go up (unless you reload, but still not cheap).