The single easiest way to get into long range shooting would be to get one of the factory offerings in either 308 or 300 Win Mag.
These two may not be the ultimate in long range shooting but they are extremely capable of making extreme range hits regularly. Also, quality brass is everywhere and cheap and they are easy to shoot as well.
Also there are more 30 cal bullets on the market then any other caliber on the market. There are also dozens of VLD and ULD bullets available in 30 cal.
The key here is to simply start learning at longer range. This is only done by actually pulling the trigger on the range.
The 308 will allow the most practice for the money spent.
I would not recommend a 338 for a "first" long range rifle. They are big, expensive to buy bullets for and generally have a big appetite for powder. They generate more recoil and muzzle blast as well.
The big 338 rounds are extremely impressive at long range but you would be better served to start your long range career with a smaller caliber.
There are no flies on the 6.5-284 either. Great round with top quality componants available.
Remember that as long as you can get close to the 3000 fps mark with a VLD or ULD bullet design you will have +1000 yard potential for sure.
Now there is a big difference between a 107 gr 6mm Sierra Mk loaded to 3000 fps and a 300 gr .338 Wildcat ULD loaded to 3000 fps. In both performance, recoil and blast.
Start out with something you can afford and comfortably shoot alot. Starting out you will learn more with high volume practice with a 308 or 6.5-284 then you will with a big 338 magnum even though the 338 will have some degree of a ballistic edge.
Remember that your skills are far more critical to shot placement at extreme range then what round you decide to choose.
For this reason I would select something in the 6.5mm to 308 cal range and look for a velocity output around 2800 to 3000 fps with the heavy VLD or ULD type bullets.
When you master that level of performance then you should invest in a larger extreme range rifle if you feel you need one.
If your going with a custom rifle
I would lean toward a 6.5-284. If you are looking at a factory rifle, a 308, 300 WSM
or a 300 Win would be hard to beat.