In the world of Shadowrun, you'll be rolling more dice than most RPG games. All of those dice will be six-sided. For the 4th edition of Shadowrun, it's probably safe to assume that you'll want somewhere between 12 and 26 dice. The exact number really depends on how heavily you managed to min-max your character and how many penalties you're taking during your actions.
The two common solutions for acquiring Shadowrun dice is to either grab a couple of 12d6 sets or one 36d6 set. As far as we know, there's no significant downside to either. The dice in a 36d6 set are smaller than standard, but buy one and you'll have all the dice you need! 2 sets of 12d6 may be two purchases, but this does offer the option of more variety in color.
Whichever route you choose, we have the dice sets for both here at Awesome Dice!