How far of shots do you think you'll have to take? .30-30 is fine to 200 yd. or so. Anything further and I'd go .308, 7mm-08, .243, .30-06, .270, .280, 8mm mauser, you get the drift. Will you be setting in a stand/blind or stalking? I ask this hecause that heavy barrelled tactical .308 will get old toating around as you hike up and down hills or mountains. Even flat land it gets old pretty quick. How good of a shot are you? I ask because if you aren't experienced, then don't take long shots. Judging by what you said, stick with 200yd or under shots so either cartridge will work fine. Actually, all the deer I've shot have been 200yd or under and I've used .257 roberts, 8mm Mauser, 7mm-08, .45-70, .30-30, and .308. I live in TX, its almost a requirement to have multiple firearms. Which ever you decide on, go to a range and get used to it. Not every gun will shoot well with every ammo. Buy some different makes and weights of ammo and shoot some 5 shot groups to see what your rifle likes best. Plus once you get the shakes from sighting in on your first deer It helps to rely on some muscle memory.
Also don't overlook other bolt actions like Savage, Ruger, or Tikka. Remingtom 700's are not the end all be all.
I wouldn't use a 5.56 or .223 for deer sized game. Another option for you. You could easily get a .300 blk upper for an AR so you can hunt with a .30 cal projectile. Just pull 2 pins, switch the uppers out to go hunt. Only thing different between the two calibers is the barrel. Mag and bcg work with the .300 blk. Uppers can be bought over the internet. The only part of the rifle that is federally controlled is the lower. Well, and the mag size for you.
Some general knowledge for you since you are new to it all. Your ar barrels are stamped with a caliber. If it is 5.56 or .223 wylde you can shoot any of the 5.56 or .223 ammo. Chances are they are 5.56. Barrel twist will help you decide on bullet weight. If your barrel happens to be stamped .223, do not shoot the 5.56 ammo. The nato specs generate more pressure than .223
Now with the .308 rifles. You are fine with .308 or 7.62x51 nato. If you get a gun that is chambered for 7.62 nato then do not, or be very careful shooting .308. Commercial .308 generates more pressure than the nato rounds. Years ago, when I was younger and dumber I had a .308 round blow out the brass in a fal learning that lesson. Luckily all I got was a shock and a little powder imbedded in my cheek.