With proper shot placement caliber becomes less of a consideration. I recommend you get a popular caliber to start with. Ammo will be more available and cheaper for practice. I recommend 30-06 or 308 as it is adequate for most big game in North America.
Hunting where? We have an old family place in Vermont. There are "hunting books" like guest books with notations that go back to the 1890s. If you sit on one stone wall by the orchard and are willing to put three days in, you will absolutely guaranteed shoot a whitetail at about 35 yards. Never, ever need more than a 30-30.
2018 deer season in Colorado I was way up in the Rockies after mulies. My self imposed limit for a shot is 250 yards. I stretched it and took my buck with a 270 Win at 280 yards. 2019, I missed my first shot on a buck at about 25 yards because I had the scope on the 270 cranked up too high and got surprised! The one I took was at about 60 yards. 30-30 would have been fine.
Deer are pretty fragile and easy to kill. It's not wanting to track them that argues for more powerful cartridges. But a good heart and lung broadside shot with anything from .243 upward will do the job. If you are hunting New England woods, a Marlin 30-30 with a good aperture sight or a vintage Weaver K3 will do you very well indeed. If you need something to reach out, 270 Win or the new wunderkind, 6.5 CM, will both be very effective and easy shooting.
You sure know how to open the large opinion can. I grew up on 30-06’s. They’ll take anything in the lower 48 and ha e the greatest range of bullet weights and types. Ammo is available no matter where you go. I have 3 30-06 rifles in different actions. Never saw a reason to get any different caliber.
Actually there is no best value gun or cartridge. It's what you shoot well enough to get the job done. I love my Savage .243 and my son loves his Ruger 30-06. Both brought home deer this year. Bullet technology, especially the Barnes solid coppers, has made lighter bullets more lethal making more calibers great for hunting like the 6.5 Grendel. And oh yes, once you move from the range to the woods and that buck steps out in front, you will have a whole new problem with you sights jumping to your heartbeat.
if you want a short fast handling levergun for whitetail out to say roughly 200 yards..........30-30
if you expect out to roughly 300 yards.......the 06 or 308
probably 98% of whitetails taken in TX are taken from 50 to 150 yards with a rifle....
a lot of those claimed long shots are measured with short steps......but every once in a while a long shot presents itself....providing you are capable of it.
Farther than 200 yards it seems is beyond a lot of folks capability accuracy-wise......a lot of folks do not know when to pass on a shot....be it beyond the capabilities of the round or especially the shooter..