One of the virtues of the Remington 700 long action is that it is long enough to accommodate full length magnums with a COAL of around 3.6". If you have your rifle fitted with a Wyatt's magazine box, you can go even longer. That gives you a lot of flexibility in cartridge choice.
You haven't said what your intended use for your rifle is, so it is difficult to narrow down the choices.
If you intend to stay with the standard .473 bolt face and/or within the 30-06 case family, you may wish to contact forum member elkaholic. He has developed improved versions of most of the 30-06 based cartridges and wildcats that are similar in performance to the old Gibbs cartridges, but with a more user friendly design.
If you want to step up to magnum cartridges, you will either need to have your bolt (and possibly the feed rails on your action) modified for the .532 case head. You can also buy an aftermarket magnum bolt, which will offer a number of other improvements as well.
The .532 case head will accommodate the standard length Win Mags (or .308 Norma), cartridges based on the .375 Ruger case, Weatherby Mags (except for the .378 based Weatherbys), H&H Mags, and RUM cartridges.
Be advised that, although Remington offers rifles chambered in .338 Lapua, most custom gunsmiths don't view the 700 action as suitable for the larger case diameter of the .416 Rigby based cartridges such a .338 Lapua, .338 Norma, and .300 Norma.
Within these parameters, you can pick the bullet diameter and case family that most interests you and best fits your goals and purposes for your rifle. From there, it is possible to make a better determination of what parts will be best suited for your application. Your cartridge choice will heavily influence things like barrel length and profile, desired stock features, desired finished rifle weight, and whether or not you would want/need a muzzle brake
. All of those things will also be influenced by your level of recoil tolerance.
When making your cartridge selection, don't forget to seriously and realistically consider whether or not availability of factory ammo is important to you and how much interest you have in hand loading. Even if you choose a cartridge with good factory ammo availability, realize that proper hand loading is still a necessity if true long range shooting is among your goals.
If you just want to accurize your rifle, you may be best served by staying with the 30-06 and going with an aftermarket stock, barrel, and trigger. Have it all trued up and properly bedded, and you might be surprised at how much performance you can actually squeeze out of the old '06 with the latest powders and bullets.