I am looking to start playing dungeons and dragons and was wondering if I can use the Dungeon masters kit with the 4th Edition players handbook and the monster manual and can I play with one set of dice as in 1 of each of the 4 sided, 6 sided, 8 sided, 10 sided, 12 sided, 20 sided, Thanks
All 4e content is inter-compatible. The DM kit works with the PHBs works with the DMG works with the Monster Manuals.
There are only a few things to be aware of.
Monster books published starting with MM3 use a different set of math than the first two monster books. This is, in practice not something to worry about very often.
Wizards published a bunch of errata for the first book set (PHB1-3, MM1-3, DMG1-2) starting with essentials. Nothing big changes, but it's good to use the updated DC tables. All of these are pretty much found in the Rules Compendium, but the rules part of the DM Kit should also contain these (it's an essentials product).
The Red Box contains a simplified set of rules and classes, it's a nice kit, but it takes some work to move past it.
That said, to address your question directly. The PHB and MM are just fine to be used with the DM Kit (and the Monster Vault if you can get your hands on it). As others have said, you can get by on one set of dice, but it's not recommended. Dice are fairly cheap as well, you can get a whole pound of mixed polyhedral dice for around $25.
I should also mention that there are three more books that need to be mentioned if you're talking about the DM Kit. That's the Rules Compendium, Heroes of The Fallen Lands and Heroes of the Forgotten Kingdoms. These books, together with the DM Kit and the first Monster Vault are the "Essentials" line of D&D products. They cost less than the typical D&D books (you can find the books for ~15) and present the most up to date version of the 4e rules set.
Lastly, I recommend this highly to anyone who is starting out with 4e, that is to go out and buy a DDI subscription. This is wizard's online service where they offer a character builder and full access to their online compendium. It's got all the latest rules and building characters is a snap. You'll save yourself a mountain of headaches for just a few dollars a month.
While I've never had a Dungeon Master's Kit in my hands, the accepted reply to this question makes me think you can use the DM's Kit in place of the DMG without any problem.
But - if I were you I'd be looking for the more up-to-date Rules Compendium instead.
As for the dice, one set is theoretically enough for everyone but as a group probably want more.
Some dice are useful to every player (the ubiquitous d20 is rolled for pretty much everything except damage) and damage dice is often rolled in multiples (several attacks deal 3d6 or 3 times the weapon dice even for a first level character) so having more than one will save you some time.
As @Dakeyras suggests, a DM might need to roll several dice at once more often than the players and having multiple dice sets could be useful. I'm not going so far as suggesting 10 sets like he does - I'm a fan of having one dice set per player and eventually collecting dice of the needed size from them when you need to roll them togheter. Since d6 are easier to find, especially if you own tabletop games like Perudo, Monopoly or Risiko!, a bag of d6s would be a nice thing to keep handy.
If you're playing a game where percentile dice are needed, some sets come with one more d10 marked from 00 to 90. If you get a single set for yourself, maybe getting one that has this second d10 in it is a good idea.