Opening day morning I always hear the big-bore bad shooters blazing away with their cannons. When you hear them empty the gun you know they missed, or at best, were shooting at a deer they just wounded.
Seen too many guys show up at the range a few days before hunting season, blast a few shells off and think their are ready. They get hung up on the size of the caliber thinking bigger is better, but then they flinch so bad they can't hit what they aim. Bigger bullets don't make up for poor shooting skills. They are more likely to cause poor shooting.
A lot of good sized deer get dropped around here with anything between .243 - 308 by hunters who practice shooting well. Some think the lowly .243 is not "big" enough for deer, but the new ballistic tip hunting bullets give very good terminal results. Recoil is very light, and excellent choices of hunting and varmint ammo are plentiful for it. Daughter just got her first deer with her .243, with a head shot at 80 yards. The Winchester 95gr ballistic silver tip never exited the deer's head, expending all it's energy for an instant kill.
Where you can put the bullet is more important than the size of the bullet.
I's recommend go to a range and ask to try shooting a few calibers your considering, before buying. Then get whatever caliber you can comfortably shoot accurately
. My daughter and Melissa both love to shoot their .243. Very comfortable so they shooter them very accurately. I'm bigger and I'm comfortable shooting my 308, which is plenty big enough for anything with hooves, horns, or claws, in the lower 48. The stepson is only 150 pounds and is barely able comfortably shoot a box of shells through the 30-06 his father gave him and he has to really work at staying accurate. With lighter calibers he's an excellent shot.
Once you find the caliber your good with, then experiment to find whichever good hunting bullet that gun shoots accurately. I've seen Winchester and Hornady ammo give excellent hunting results. So,those would be a good starting place to see what your gun likes best.
Then practice proper shooting technique until you can put all your shots into a fist sized area at the distances, and under the same conditions, that you'll be hunting in. Don't settle for the often recommended "paper plate' sized kill zone of deer. In real world conditions, you want to be better than the minimum standard that some hunters are willing to accept when at a shooting range.
Then, well before the season opens, start spending as much time as you can scouting and learning what the critters are doing in the area your going to hunt. As hunting season approaches, get to know the critter's living room, not yours.