First thing that you're aiming for is getting a Sattelite Nexus or whatever it's called up. (the building to give you more satellites). Also, build your other satellite right away and launch that to a place that is panicking. Of course, you will need power and space, as well as engineers to make the satellite relay thing, so you gotta do what you gotta do to get them up, which would mean building a power generator quickly, getting engineers (so try to do a mission that rewards with engineers right away if you can), and having the cash. You probably wont get the satellite thing up until the second month at least.
Even later in the game, keep it a priority to keep up on satellites, try not to have any countries pull out of the program (though a few might and is probably unavoidable).
Also, don't send a satellite up in a place with no interceptors, at least get one there right away. Also make it a priority to get the upgraded interceptors (the alien looking ones, think they're called firestorm or something, forget it exactly) and equip them with phoenix cannons or something, because later on you will get ships that will demolish those crappy interceptors in like 1 shot.
In missions, try to have at least one of every class in your team. Early missions, try to have at least 2 of the classes present, but as said above don't just stick to one team. I usually have 3 people with classes and a rookie for early on at all times, trying to get all the rookies upgraded. Rookies can be better if you give them a scope early on, just for a more useful soldier in battle.
Then think about the context of the mission, and what classes to bring. Like for crashed ships, don't bring too many (if any) heavy soldiers, as you don't want to destroy the ship's engines and whatnot (worth money and for upgrading stuff). The all out destruction levels (the ones where you're basically forced to go to a country cus the panic went too high), is a good time for heavy soldiers, as you can blow anything up. Snipers are good for those rescue levels, and basically any open levels. When you got a bigger squad, in lots of missions I like taking 2 snipers, one with the upgrade where he can shoot people that are in the sight of allies, and the other sniper with the snap shot upgrade making it so they can shoot after moving (with a small penalty).
Now in battle, try not to have anyone "dashing" if you can help it (of course, guys that are behind should usually have no problems dashing to catch up). Just move their short movement increments a turn, overwatching or hunkering down at every turn. The worst is when you've moved all of your team mates, and then the last guy runs intoa group of 3 guys, almost assuring his death the next turn. So move everyone, and before overwatching, make sure everyone has moved (incase one of the later guys run into someone, you can still shoot with your other dudes) then overwatch with everyone after no one can move without dashing.
Try to keep at least one person (sometimes better to have more) with a medipack, so you can heal poisons or just health if your guys are damaged too much.
And last but not least, have some fun and customize your team. I like changing the looks and names of my dudes to make them all my favourite over the top action heroes. So ill have a team of Arnold Schwarzeneggar, Bruce Willis, Sylvester Stallone, Jean Claude Van Damm, Harrison Ford and Mr. T.