In the lower price points what you lose out on is setup from the factory, and QC on the electronics. So you should expect to have to check/adjust the truss rod, action, nut height, and intonation. Easy to deal with, and If you bring it to a local shop/luthier could cost anywhere from $50-100. Other things might include fret level and smooth fret ends. Those are more costly to deal with. If you can go to a local shop to get your hands on instruments, that's going to be the best thing for you. See if any of the fret ends scrape your hand/fingers/thumb as you slide your hand along the neck. Make sure the neck and fretboard feel good in your hand.
An Epiphone SG can get into your price range and the QC on Epiphone, especially the Inspired By line, has been really good lately. And an SG, especially if you later upgrade the pickups to have coil tap or coil split, can be pretty versatile.