I have served in 2 of the 3 branches and worked with medical in many joint environments. I will give my 2 cents.
1) ARMY: best chance of training without a GMO tour (the best idea is to avoid gmo). Army also has all the specialties that you may desire. For someone not wanting to train in Primary Care, the army is head and shoulders above the other two branches. (they are so much bigger that they can train more specialists). They do deploy for longer than other branches. (for some this is a major draw back) I know many more happy Army docs than any other service.
2) Air Force: They still are the most laid back (but keep in mind, it is still the military). They primarily have primary care positions. (anything else requires you to be prior service with a great board score. Not good but great). Don't be dismayed, their GMO tours can be awesome. You can fly in jets and planes as a doctor. (flight surgeon). Read more about flight surgeon in any air force recruiting page. Know that to get a competetive residency you may be required to do a GMO tour. (also same for NAVY)
3) Navy: Very tight knit group of physicians. Also primary care oriented. However, navy is the highest GMO producing branch in the military. Rumors have circulated for years but this program still exists and will for the future. This is a source of many unhappy navy physicians. This branch seems to have people that love it and people that hate the it. Not many in between. Also, you will spend time on a boat if you get a sweet base near the beach.
Hope it helps. that is my experience.