I've killed 70+ deer (typically at 150-250 yards and up to 200 pounds) with a .223 firing 45 grain jacketed hollow points at 3500 fps. They fall in their tracks.
Since deer can't read all the marketing hype about various cartridges, they just simply die when hit. I've actually seen a deer that was accidentally killed (not by me) by a pellet gun from 40 yards. So, any of your 3 caliber choices will be more than adequate for your intended targets.
My favorite of the 3 is 6.5x55. It's super cheap to shoot- you can get 20 round boxes of 139gr soft point Prvi Partizan ammo from
www.aimsurplus.com for $13.95. I've killed deer and groundhogs with these bullets and they all were quick kills. I get 1" groups at 100 yards with all 3 of my 6.5x55 rifles with this ammo, too.
Handloaded 85gr 6.5mm Sierra hollow points at 3400fps are truly amazing on groundhogs. I launched one 3 feet in the air with a shoulder shot from 50 yards when it was hunkered down on it's belly trying to hide from me. The shock from these bullets consistently blows the guts out of the groundhog- even when you hit'em in the shoulder! Amazing. And the deer I've shot with these look like a bomb went off inside them. Bang, flop.
A 140gr 6.5x55 has recoil energy equal to a 100gr .243 according to Chuck Hawk's website-
http://www.chuckhawks.com/recoil_table.htm . So it's comfortable to shoot, even with a much heavier bullet.
And, you can pick up a sporterized Mauser 96 in 6.5x55 very reasonably. Mine's probably worth $400 or so with it's Timney trigger- not bad for a rifle that's MOA accurate with $13.95 a box ammo and will kill anything on our continent!
That's just my personal preference, of course. Borrow different rifles in different calibers from friends and try them out at the range, then shoot some critters with them. See what they do to animals of various sizes with different bullet constructions and velocities if you have the opportunity. I'm lucky to have nuisance permits to shoot deer, so I can experiment and see what happens in the real world. Best of luck and whatever you do, get out and enjoy your new rifle!