“What would be a good variant for someone on a tight budget? Caliber, etc?”
My advice is to slow down and do a bit of study before making your purchase. Type in ‘AR-15’ on YouTube, and watch videos for a few days before you shop.
At their heart, these rifles are simple, but there are quite a few key areas of mechanical differences that you should familiarize yourself with so that you’re happy with your choice in the end.
I would start with a Ruger 10-22 with some 25 cartridge magazines. These are the most under appreciated weapons made. Many a GI was killed with 22 cal sniper rifles in Germany. There are some YouTube videos showing that a 22LR can be fatal at 300 yards. In any case it is a cheap way to get familiar with shooting semi-auto.
I have a love affair with the Ruger Mini 14 since it is very similar to the rolling block M1 Carbine and Garand of my youth. Not cheap though, over a grand on their web site. You can probably pick up a M1 Carbine cheaper but the ammo is ridiculous, likely to pay 70 cents to a dollar a round. The 223 is just a hopped up 22.
Good Gun
Bad Gun
Same Gun Different Stock. Ruger Mini-14