This is for Classic Ironman. No idea if it would work on Impossible.
Start in Asia to lower costs.
Initial priorities: beam weapons, carapace armour, satellite coverage. Ignore everything that doesn't advance those things. Get satellites over Russia, USA and all of Africa as soon as possible (for the most funding). South America has lowest priority (hardly any money and their bonus sucks.)
My squad is usually two support (medic build), one of each other class and one SHIV. I treat the SHIV as expendable. It's gets sent into rooms first and if someone else is in trouble I'll drive it out into the open next to the aliens to draw fire. The other soldiers operate as you'd expect. Heavy handles crowds and mechanical enemies (yay HEAT ammo). Sniper gets up high (skeleton suit, elevation bonus, squad sight) and picks shit off. Assault holds doorways (close quarter specialist), draws reaction fire (lightening reflexes) and delivers the killing blow on anything the sniper can't kill.
I'd be interested to hear other people's preferred soldier builds. I've got Assault and Support all figured out, but I can never decide whether squad sight or snap shot is better for Snipers. Same with holo targeting and bullet storm for Heavies. Holo targeting is more useful most of the time, but when you open a door and find yourself looking up a Sectopod's nostrils, you'll quickly come to appreciate the wonders of bullet storm.