I only have three barrels at this time, so I am not an expert.
The .243 was the reason I bought my first Handi. I wanted a .243, but had no real use for it. So since the Handi was cheap, I bought one. I mostly got it to shoot, through a lot of hand loading and trying things from the FAQ. That is OK since experimenting is part of my attraction to the hobby.
Since barrels are cheap, I got two more. The Ruger .204 was no fun. It shot MOA with the first load I tried. The .500 S&W took a couple of tries, but was not much of a challenge.
I have heard others say that the .243 is difficult. If you are not a reloader, maybe the 25-06 would be a better choice. I have not heard any bad things about it.
mike