If you want to learn to shoot long and also start to handload, forget your first three choices. They will never keep up with a 6mm or 6.5mm at anything over 500yds. The bigger rounds just carry more bullet weight and shoot bullets with much higher BC. The .223 lacks the boiler room for the new heavy VLD's in my opinion, and it needs to be loaded to excessive pressure in a long barrel with a super fast twist to get the velocity needed. And from what I have heard, it's not the easiest route for a beginner. You can always use lighter bullets, but that again will just put you at a disadvantage compared to the bigger rounds.
The 22-250 is a bit of a barrel burner, and the 1-12 twist is far from ideal for the heavier bullets, which is what you want if you plan to shoot long. The .204, while accurate, also lacks the ability to throw heavy bullets. The .223 has a bit better twist rate at 1-9, but it still limits you to bullets of about 65 grains, maybe 70. While the 80 and 90gr bullets, which have the high BC and weight you need for the long game, will still be out of reach for the 1-9 twist.
I think a far better choice to learn to shoot at long range would be one of the Target Rifle series in 6mmBR with a 1-8 twist, either the Model 12 Bench Rest or the Model 12 F-Class. They both have nice wide flat fore-ends made to ride the bag in a front rest, and have an ideal twist rate for the heavy VLD's that will really buck the wind as the distance starts to stretch. The 6BR is also a very easy round to load for, and top notch components are easily available. It will easily out-shoot any of your other choices, especially at longer range. It also has the capability to shoot accurately out to 1000yds in good conditions, something the others will struggle with.
If you really like the LRPV rifles, the Left Port has the option of a 6BR with a 1-8 twist as well.
The 6BR is a much better choice for long range shooting and is very easy to load for. It's easy on barrels, if you take care of it and don't abuse it or clean it to death it could last 3000-4000 rounds, about 2 times longer than a 22-250, and be more accurate in the process. It's one of the best rounds going for shooting out to 600yds, and would be a great choice for a beginner as it's very easy to tune. I just don't think there is a better choice to be had anywhere in any rifle for a beginner learning to shoot long and reload as well.
Another good choice would be the .308 Winchester. It has good factory ammo available, and has plenty of power to reach out accurately. It will cost a bit more to load for, and also will have more recoil as well. Those are things you need to consider. But the barrel will last almost forever, as some have gone to 5000 rounds still shooting well.