For varmint shooting it's hard to top 223 for availability,efficacy,cost,etc. There are other .224" center fire cartridges hotter (22-250,22 Swift,etc) they "out perform" the 223 but for utility sake not by huge margins. The sub .224s like 204 Ruger and the 17 center fires work well,just bullets are light and you don't gain much advantage over 223 again.
For deer/antelope sized game you can start with 243 Win/6mm Rem performance with quality bullets at 95 or 100 grains,with proper placement and terminal performance,but if I had to pick a single "deer" cartridge it would be 270 with preferably 150 grains if they shoot well in the specific rifle. Now,if you hunt deer and elk,and maybe moose or bear then you start with 270 then include 7mm Mauser,7mm Rem Mag,280,308,30-06 and a few more in that range.30-30 is a capable cartridge within it's range,especially with the LeveRevolution ammo out there but it's still a 150 yard cartridge.
For big game you need to start with 375 H&H and 458 Lott isn't too large for some things.
Personally I think 30-06 is a wonderful cartridge,gives better performance than 308 Win in equal bullet weight and has the advantage of shooting 180 for 200 grain bullets with no problems while the shorter neck on the 308 case can cause powder impingement issues at SAMMI COL specs. With the premium bullets I load,I have no hesitation using 30-06 for anything but dangerous game/big game. I have a 300 WSM also,not near as pleasant to shoot as 30-06 and it doesn't do anything the 30-06 can't within realistic hunting ranges,but it was a gift so can't complain. 30-06 is available in wide ranges of weights and performance also.
I like my Browning A-Bolts,they are handsome rifles,they have a short bolt throw,removable magazines and I get sub-MOA accuracy easy enough.To me the A-Bolt original and 2nd series are great,not so happy with the "economy" one out there now. The X-Bolt is a fine rifle,it has the upgraded safety system and trigger. I like the CZ 550 American,Ruger M-77,Tikka T3 and Sako offerings too. I grew up shooting a Rem 725 BDL Custom, but current Rems are not something I'm enamored with,I can easily find a better rifle at most price points unless the Rem is a super sweet deal. Entry level Savage and Ruger rifles,the Axis and American are real values for the price,the Weatherby Vanguard Series II is maybe best value in a rifle currently.