Generally, most people upgrade the Vanguard in the following order:
1) Level up Charge to Heavy Charge
2) Level up your passive to Champion/Destroyer (your choice).
3) Level up Incendiary Ammo to Inferno Ammo.
After that, it sort of depends on how you want to play your Vanguard:
1) If you want to continue a charge-heavy playthrough, then level up Cryo Ammo to Squad Ammo. This will give your squad a decent ammo. After that, you can sink your points into whatever you want, either Shockwave or a bonus power.
2) If you want to be more caster-oriented and not charge as often, then put points into Shockwave and Pull.
If you are going to charge, then shotguns are king. I like the Katana, although not everyone does. You can get the Scimitar later which does less damage but has a much faster rate of fire.
Note: Pistols and sniper rifles deal extra damage to armor. Shotguns and SMGs deal extra damage to shields and barriers. Assault rifles deal a little extra damage to all defenses. All weapons except sniper rifles deal more damage the closer you get to the enemy.
Later in the game, you will get a choice between a new shotgun (the Claymore), sniper rifle training, or assault rifle training. I recommend the Claymore or sniper rifles on the Vanguard.
As for upgrades, you will find upgrades as you progress. Each weapon, has 5 damage upgrades by default (no DLC; some DLC provides extra damage upgrades). Same with some other upgrades like Omni Tool and Biotic Amps, and Medigel and Damage Protection. For each of these upgrades, the 2nd and 3rd upgrades will unlock special upgrades unique to the weapon. Shotguns can get more shield+barrier damage and spare ammo.
Vanguard does not care that much about squadmates, although a few can be good. Miranda (and later Thane) can be good with Warp if you use Pull first to create a biotic explosion. Kasumi (DLC) has Flashbang which can be good for disabling enemies first before you charge in.
And finally, be sure to check out some of the pinned threads in the ME2 Classes and Builds forum (they are the ones marked Pinned at the top of the forum) as they contain some useful information, links, and build ideas to newcomers.