I only started shooting F Class this year so I am not professing any huge amount of experience. But, being one small step ahead of where you are at now, hopefully some of my research and reasoning may help your quest.
First off, F/TR is a great place to start, .223 and .308 compete side by side and against each other.
I started with a .223. The main reasons being it is easy on the wallet and easy on the shoulder.
Not having a huge amount of experience shooting past 200 yds. I want to practice, practice, practice. The .223 allows me to do this quite nicely, as it is roughly half the cost of .308 for reloading components. Also, with reduced recoil vs. the .308 I can shoot the .223 for as long as my concentration will allow, and not worry about getting a sore shoulder or developing bad habits due to recoil.
The drawbacks to the .223 would be that it is very sensitive to variations in the powder charge. If you don't reload, I understand match ammo is more difficult to find for the .223 than the .308. At longer ranges the .223 will be more susceptible to wind drift than the .308.
If you do reload and pick the .223, get a scale with high sensitivity and pay attention to your charge weights. After talking with a couple of very knowledgeable people who reload the .223 I try my utmost to load to a variance of less than .1 gr.
The .223 and .308 are very similar ballistically, but obviously if Mr Newton is correct, the wind will play more havoc with an 80 gr. .223 than a 155 gr .308 travelling at similar speeds. At the distances I shoot ( mostly 300 yds., but out to 600 meters when I can) this is pretty much a moot point. From my understanding 900 yds or more is when the smaller, lighter .223 is at its greatest disadvantage in the wind.
If you do decide to go with a .223 you will best be served by finding a rifle barrelled with 1in8 or faster twist. This will allow you to shoot the 80 gr. bullets that do best in the wind.
Soooo,,, the .308 while possibly outperforming the .223 at longer range on windy days was won over for me by the fact that I can practice more and longer with my .223. I don't feel the caliber has handicapped me in any way. There are some very successful .223 shooters where I shoot and they consistently take home hardware at the end of the shoots they participate in.
What ever you choose I wish you luck, get out, burn some powder and have some fun.