With a LA Savage, pretty much any round you wanna make is doable. If you go from the .532 to the .473 boltface this opens a whole slew of effective deer calibers. It comes down to what do you want out of this round, how much time, effort and money you want to put in to it.
Since your shooting skills are "basic" and a round that is easy to shoot and great to learn off of is the .308Win. Great round and plenty of data and components for reloading. Plenty of factory fodder and with basic skills the only 308 family round I would remotely suggest out to 800yds and even it's "iffy". Yes the .243, .260 and 7mm08 will kill deer that far but it takes precise shot placement for effective, humane kills. All of these rounds are "running outta gas" at this distance.
Any of the .30-06 family of cartridges would also be good choices based on the above criteria. Again, I feel the .25-06 would be marginal, and only the 115gn bullets should be used. I've never slung a 115gn Berger that far so I can't steer you either way. At closer distances, 100/200yds, that are the Hammer of Thor! Very impressive on whitetails.
There are wildcats and AI versions of almost everyone of these two family of cartridges. Powder cosumption (i.e. barrel life) is 50gns or less and pretty ecomical. Recoil tolerable for most folks and accuracy is great, most of the time.
With the bolt head that you have you can easily stay with the WSM's or original 1960's "short magnums". Any of these would and could easily take deer out to 800yds and in some cases, beyond. A little more powder and a little heavier bullets come in to play here and thus your recoil and powder consumption goes up.
The 6.5WSM (wildcat) is a great round, so I've heard. Brass would be available, but I'm not sure about barrel life. The other three or four WSM's (.270, 7mm, .300 & 325) would make good deer rounds also. I have owned the .270WSM and it is a superb round. The other three I can't speak for on taking game. Others on here swear by them.
The family of belted magnums are an effective group of cartridges. These require a little more powder, recoil can be stiff, but the accuracy is there. They carry the payload the distance but you pay.
From there on up you have the RUM's, proprietary lines (i.e., Weatherby, Lazzeroni, Dakota and Allen Magnums, etc.), the para-military (.338Lapua, 408CheyTac and 50BMG). There are not efficient to shoot but are effective when they hit, if you can bear the recoil. LOL. With muzzle breaks and ear protection these are manageable.
You have numerous choices. It really comes down to personal taste. Good Luck. JohnnyK.
