I can provide a few answers, as they mention them during the series, at least in the dubs.
Missing from here is Captain Avatar. The Captain's wear the peacoat looking uniforms. On the far left in the green is the Fleet Commander in charge of Earth Defense Command. All Earthbound EDF/EDC members wear the green uniforms.
White with Blue usually indicates individuals in the Sciences divisions. Sandor is referred to as 'Chief', so it is assumable that he is the head of the Sciences division.
White with Green is the 'Tactical' division. Navigation (Mark Venture leads this division). This also includes communications and tactical, including Nova/Yuki and Homer (communications officer), who both at times sport the Yellow and Black uniform.
White with Orange is 'Environmental'. This includes Engineering personnel primarily, lead by Chief Orion. Orion is the only member of Avatar's Command Ship crew from the failed battle at Pluto to serve aboard Yamato/Argo. It is he that helps Wildstar come to the realization of what truly happened at Pluto, and the fact Avatar also lost his daughter in that battle.
White with Red Cross is, obviously, medical personnel. This included Mimi the Rescue Cat, Dr. Sane (with his omni-present sake/"spring water") and when serving in her Nurse capacitiy, Nova/Yuki.
Black with Yellow would eventually become the signature of the Cosmos/Black Tigers squadron. Early in the series most of the pilots aside from Conroy ('Second in command of your fighters' he intros himself in the early eps) sported the White and Red of Command. Later in the series, all pilots other than Wildstar sported the proud Black and Gold of the Cosmos Tigers.
White and Red is Command and Control. Wildstar, as the gunner, deputy captain and eventually acting captain of Yamato, wore this uniform. Dash also did, serving as Wildstar's "wingman" on landing/away missions on multiple occasions.
As for Actual Ranks, from what I remember, only Captain Avatar, Lieutenant Derrik Wildstar (eventually Deputy Captain) and Lieutenant Mark Venture were given formal ranks among the Star Force itself. (I hope I remember right, lol). These ranks were given to Wildstar and Venture in the first couple of episodes when they recovered the message from the downed ship from Iscandar while serving on the Mars outpost.
The uniforms do not contain a specific rank insignia in the original show. This is due to the fact that this was supposed to be an assembly of individuals, the best from the World over. Command structure was left up to Captain Avatar. This was explained in that the Earth unified its space fleets only in the face of the Planet Bombs and Gamelon/Gamilus attacks. Prior to that, the nations conducted their own space operations. There was little time to formalize a military structure involving ranks.
I hope that helps and is found informative. :)