The Savage 10/110 is a good bolt action to learn on. They have a solid reputation for accuracy. There are several aftermarket accessories available to customize it. And when you get bored with it, or if the barrel is worn out, you can change the barrel yourself at home. A new barrel is less $ than a new rifle and you already know the trigger and action. If budget is more of a priority the Savage Axis has the same list of benefits. It takes the same replacement barrels as the 10/110. The biggest issue with the Axis is the flimsy stock they come with. Some people have reinforced the stock with epoxy. I bought a replacement stock from Boyd's Gunstocks. Some people don't like the looks of a Savage, but beauty is in the eye of the beholder.
As far as cartridge goes, your choice of 243, 7mm-08, or 308. Cost is similar. Recoil can be heavier with the larger bullet diameter. But all are good deer cartridges. My personal lightest for elk is 7-08, but many hunt elk with 243 successfully. All 3 punch holes in paper quite well.