Everything except armored heist and big bank unless you plan on hosting them. From most essential to least essential:
Gage Pack #1
Gage Sniper Pack
Gage Mod pack
Gage Pack #2
Armored Transport
Big Bank
I beg to differ.
Armored Transport should be the first to grab because they give easy cash if you get the Train heist, and the 3 weapons are all really good. Gruber Kruz and Swedish K are IMHO the best stealth guns. Gage Pack #1 should be an instant grab next because grenades and the Spec Ops. Some argue that the Eagle Heavy Rifle is good enough to compete with the Car-4, but it's your choice. Depending on how you play, you may want Sniper Pack or Pack #2 for either sniper rifles or LMGs respectively. The last knife you unlock in Pack #2 is useful for stealth due to it's extra range though. Gage Mod Pack has lodsauseful weapon mods, but you'll want the weapons first. They're mainly weapon sights. Big Bank is the newest one, but it includes pre-planning heists and obviously the biggest map in Payday 2.
It depends highly on whether or not you plan on hosting though personally I don't really like the train heist weapons. Swedish K is good though.
If you can though, it's probably just best to grab all of them except maybe big bank.