Some general tactics, mostly obvious:
1) Focus fire. When new obstacles are revealed, formulate a plan of attack that involves everyone working together.
2) All about the Nuyen. Learn to love those 8-11 nuyen obstacles, they give you the cards to solve the next problem. Know when someone is just shy the amount they need to solve a problem (Covering Fire, Doc Wagon, any of the 9 costs, etc).
3) Assist! Those assist cards have a serious advantage in that you can use them a round faster to prevent damage or clear a crossfire.
4) Martyr. If one person gets staggered, even every round, you can use healing cards to get around it and eventually recover. If more than one person staggers at once, the end is nigh. The Street Sam has salvaged multiple games by whipping (or strikering) things over so that the rest of the team can stay alive until they're whittled down. You just need 1 heal per round to pull it off, Crossfire willing. (Disclaimer: Don't overdo it. It's just one tool in the kit)
5) Don't Hold Back. Conserving cards and nuyen for a plan down the road can have great dividends, but the crossfire deck can also screw up everything. At the start of each round get a feel for what the table can take out, then execute. Do anything to keep the crossfire number down. It sucks to start the next scene with hardly any cards, but it sucks more to have a higher crossfire number and take an extra round of damage.
But, really, at the start of each round / notable change in game have a brief discussion and stick to a plan. If everyone willy nilly picks away at what's in front of them, your crossfire # will climb, obstacles will rack up damage and possibly even start healing back.