Thoughts on team games (because I've been playing random 3s so much lately):
1- People tend to leave too early and react poorly once a person is rushed.
In a 3v3, once someone is attacked, they usually call for help right away while trying to push them out.
There's an art to helping someone. If you help too early, your army will get wiped out by three, especially if they decide to stream in reinforcements. Also, the defender usually makes the mistake of trying to engage in a lost battle before asking for help. That's not the right response... if you can't win at the moment, you need to rally troops to the back of your base and try to delay them hitting the army for as long as possible. That way you can attack with as many as troops as you can once it becomes desperate and inflict as much damage as possible. If possible, attempt to delay them until your teammates come; naturally, they'll be surrounded and will die.
Let's say that one of your partners gets completely taken out... you actually have an advantage, but if you press too early you'll lose it. Don't rush out right away and counter with a conventional attack; either systematically take out their remaining army after your partner died, or wait about 2 minutes and claim your small tech advantage and/or superior army size. When you attack right away, you won't do enough damage to take them out because your tech or size advantage hasn't manifested yet, and their backup troops are usually enough to prevent you from doing significant damage.
The exception is if your tech advantage is a harass they can't counter (DTs/Mutas are usually the main ones, drops might work too), in that case, you want to press them immediately with it.
Bottom Line: Don't attack if you know you can't win, and unless you have uncounterable stuff, wait a few minutes before you counter.
2: The bigger the team game, the less important harassing their economy is.
Harassing is more effective if they have multiple expansions, because you want to catch their army out of position. This rarely happens in team games, because 1-base play is the norm. In addition, denting one person's economy by a few workers doesn't have the same effect. In 1v1, you take out 6 workers, you're up 6 workers. In 3v3, you take out 6 workers, that still means that you're tied with the other 2 economies on the other team. Unless you do some insane play like double dropping the same guy with blue flame hellions (and challenging his ability to multitask), tiny damage doesn't add up in big team games.
There are exceptions of course... like if they're obviously not prepared. If you can read your opponent and you know he can't defend mutas / DTs / small drops, then go right ahead... but generally the amount of money you put into your harassment costs more than the damage done in a big team game.
Bottom Line: Small harass isn't as effective as it is in 1v1, and it grows with more people. Harass needs to be devastating in order for it to be effective (either 10+ workers or destruction of Nexus/Hatch/CC).
3: If there isn't an early rush, the most important battles come with every expansion.
The expansion is the most important timing in a team game. While I said that harass isn't important, what IS important is completely wiping out expansions, which can be done easily. One person can send their army there and clean it up, maybe a drop or something. And once you displace their army while that player helps himself, you can ambush their reinforcements with the other players. Having the ability to hit and run is important, because if you can't take out their base, you want to at least move the opponent's army around to see what they have or to catch them out of position.
With that said, prepare accordingly. If you can take an expansion and are successful in defending it, it's a won game. Don't let your teammates lose their expansions carelessly and work tirelessly to deny them theirs. Depending on the map, the key battle might be the one for the first expansion or the second one.
Also, don't ignore the second expansion. A lot of people aren't aggressive with expanding and it usually costs people in the game. It's also easier to hold expansions earlier in the game... towards the end when everyone's desperate for resources it becomes a fierce battle for the few remaining expansions, but by expanding relatively quickly and ahead of typical expansion timing, it's easier to hold it.
Bottom Line: Games are usually decided by battles for expansions. Therefore, always try to take and defend them, and deny them when you can. The smaller the force you use to take out an expansion, the better.