Primarily target shooting. I dont have much experience with different chamberings. I have shot a few, but it's an embarrassingly small sample size. My thought was to start varying my collection (I.e., have an AR, what else should I have?). This will be my first bolt gun.
I am so pleased with this ruger ranch 5.56/.223 that its sister in 300 blk is on her way to the FFL....
.223 would be cool, especially since you have an AR.
Most of your "traditional" bolt action rifle calibers aren't cheap to shoot. Unless you stock up on surplus .30-06 or you reload, you're going to be spending around $20 a box of 20. I say around because you can find cheaper ammo like PPU but you can also find more expensive ammo like $40 hunting rounds. Remington Core Lokt is a happy medium price wise. A magnum rifle like a .300 Win Mag is going to be similar if not more. Again, assuming you don't reload. The .308 is the standard of tactical bolt actions and I can tell you that except for imported stuff like ZQ (do they even sell that anymore, considering politics in Turkey now?) and PPU, the .308 is not cheap to shoot unless you reload or buy surplus 7.62x51.
That being said, I would look for a cheap caliber for a first bolt action rifle. I would look into .223, 7.62x39, 7.62x54r (I know, I know, OP didn't say Mosin) and maybe 7.92x57, which could be done if you take a beat up .30-06 Savage and get an 8mm barrel for it since they share a similar sized bolt face.
I find 7.62x39 fun to shoot myself and Hickok45 had no issues with the CZ he shot. Savage at one time made a 7.62x39 rifle but it was a niche model. Ruger also made one.
Exception to the rule would be .243. It wouldn't be cheap, but it is plentiful at gun shops around the country, wouldn't run out in the case of an ammo run (Hillary '16) and is a good starter cartridge for young hunters-to-be. Or young-shooters-to-be.