You do need other sizes of dice, but you could use a free dice roller app on a laptop/tablet/phone for now.
I assume you're talking about 5th Edition D&D, which unfortunately doesn't have a lot of material for it at the moment. And certainly not much free stuff. Paizo makes some cool Free RPG Day Adventures, but those are for Pathfinder so would require a little work to adapt them to D&D.
I can say the way I learned to GM was by running modules. But there aren't very many free ones for D&D5 to get started with yet. You might want to get your group to all pitch in $5 toward Hoard of the Dragon Queen, but I hear it starts out rather tough. The adventure throws the group into a chaotic situation, and doesn't expect them to have time to solve all the problems going on at once. So remember to let them retreat and rest, even if it means they "fail" at some of the objectives.
The important thing is to not worry too much about getting things wrong. It's ok to make mistakes, just go with the flow and try to let everyone in the group shine at least once a session.
My word of advice: Play a game of D&D before dungeon mastering, it helps an awful lot. Find a local game store that hosts D&D games, create a character, and play through a session. It is difficult to be a dungeon master if you haven't played the game or know how things will play out in the long run of a campaign, plus you will often times meet experienced players or DMs who can offer you words of wisdom.
For dice I'd suggest finding a store that sells them in bulk and buying the individual dice you need. It's much cheaper than buying a set of dice in a box.
As for dungeon mastering don't worry if you make mistakes or come up short. That's the beauty of GMing and you'll only get more creative the more you play D&D. I'd suggest going the route of playing a pre-made modules like deinol said, as they are far less time-consuming than running a game of your own design.