Having just finished my Commander playthrough, I thought I might offer my insights to help everybody else, as this game can be challenging and frustrating on this difficulty.
HERE BE SPOILERS!
1. If you are getting frustrated, don't give up - it gets easier. At the beginning, you have your shitty M14 rifle and no skills, and the Outsider soldiers (the basic enemy) shred you. As you level up and gain skills (and especially find some laser/plasma weapons) it gets easier. I found the University (the first Major Operation) to actually be the most difficult mission barring the final boss arena fight). The basic Outsiders stay the same and the crowds get much more manageable. Checkpoints are fast and furious in this game - before and after each arena fight), so don't worry about failing arenas - learn the layout and spawns and plan for the next time.
2. Almost all fights are arena-battles. Apart from two instances in the last op, you will only ever have Outsider grunts (with the possibility of some commander types) and potentially one (and only one at a time) heavy enemy near the end (Muton/Sectopods). Regarding target priorities, you absolutely must kill Drones first - they heal other enemies, lift your guys (where they will get shredded) and fly over cover to flank you. Next up are 'Commander' type enemies (alpha with a Lift/Crit Strike and shooting, see below), especially 'Minion' (spawns drones) and 'Psion' (splits himself and Mind Controls your guys - bad times). Third are Phantoms (Cloaky flanker guys) and Outsider Elites (heavy rush guys). When heavies spawn, they move to the top of the table, barring drones. Use the alpha tips below.
3. Regarding team composition: I found a recon and engineer to be the best team. Their abilities synergize very well with your own. All of Recon's abilities are useful. First is Critical Strike (to quickly kill commander-type enemies - I found lifting lifting them with your power, calling in a CS and then shooting them to be a good way to quickly kill the most dangerous enemies early on). Second is Fire Mission (lift a Muton, call in a FM and hit him with everything you've got). Third is Distraction - if you get the add-on perk for it, this distraction can last for a very long time and draws _all_ fire - great when you start to get crowded. An excellent way to Alpha a Muton or Sectopod is Lift, Fire Mission, Mine (directly beneath them) and blaster launcher shots. You can kill them in one swoop this way and this is very good in later stages.
The Engineer has the Mine (which is great for keeping you from getting flanked, and if you fail a checkpoint, you can place mines all around before triggering the switch or whatever triggers the spawn), and the all important Turret. I recommend the Laser Turret as it is mainly a distraction for when you start to once again, get crowded. I did not use scatter much, but it can be very good if there are a few tough enemies to flush out. Also, when you tend to get rushed on the later stages by Outsider Elites, he has a chance to kill them first with his shotgun-type weapon. Support were just too squishy and their shields get overloaded too quickly (despite the awesomeness of combat stims) and on Commander, the real damage is from you and abilities, not your teammates guns, so Commandos were not particularity effective IMO.
4. Regarding weapons: I found the best combo (for me) to be the M14/Machine Pistol, followed by Laser SMG/M14, followed by Laser Burst Rifle/Blaster Launcher, followed by Plasma Assault Cannon/Lightning Gun. Once you get the Blaster Launcher (an RPG weapon, keep it till the end level where the LG is - you _need_ a heavy weapon for the Mutons/Sectopods/Gunships). In general, I found shotgun weapons bad as you take so much damage so fast that engaging the enemy at close range is very likely to end up with you having to reload the checkpoint.
5. Regarding your own skills: Upgraded lift is crucial for killing Mutons/Sectopods later on, and for immobilizing tough enemies quickly to give you breathing room to run away. Make sure to get the combat drone as you lose health too fast for the heal drone to matter. Keep the blob/drone up at all times to keep the enemy distracted. Once you get Mind Control, make sure to get the damage/health buff - you will love it when you MC a Muton Elite who proceeds to kill most of the wave by himself. A quick note regarding Mind Controlling heavies - only do so if they are in the open nearby other enemies - otherwise they will spend their entire MC period walking slowly towards other enemies and not killing. If they are in a bad spot for MC, the next best targets are Phantoms/Phantom Elites as they are close-in specialists and will usually get a kill or two before getting killed. Don't be afraid to spam heal - your agents tend to lose almost all of their health very quickly, and you often cannot expose yourself to revive them.
6. Regarding positioning, I found it impossible to set up flanking maneuvers and such - my guy would get one Outsider and then be alone too far forward and get shredded. Stay back at the arena entrances (which tend to be quite chokepointy), mine up and whittle them down. In general the fights are over before you run out of ammo, and there is always ammo pickups before the next arena.
7. While it may be tedious, make sure to do all the Minor Ops - you need the XP to get to level 9/10 and get that all important MC buff (and cooldown reduction which really helps on the last op). Make sure to do all the dispatch missions to level up your other operatives to lvl 5 in case you lose your guys - they need little input and are a good insurance policy. However, in general, if one of your guys goes down in an arena, you should restart the checkpoint, as it is very difficult to beat some of the fights with one or two guys down (the Engineer is okay to go down sometimes, but I found the Recon agent to be quite critical).
8. Keep at it - the fights can get frustrating, but all you need is one good MC on a Muton/Sectopod or 5 minutes of setting up mines to prevail - you can do it!
If I have missed anything (or you have an insight), please reply in the comments and I will update the solution - good luck!