Keep in mind that Red Box is D&D Essentials and to really build a character using it (like, actual customization, it does come with pre-generated characters), you would need the Essentials handbook.
I just started playing six months ago and this is what I find useful:
1. Dice, I would recommend a set for each player but you technically just need one set
2. A DM and an adventure to play. You can either create your own (not recommended if you've never played). I would recommend picking up an adventure book since it comes with most of what you need and does most of the work for you.
3. A map/hex grid. Again, if you buy a pre-made adventure, this is taken care of.
4. Minis/Tokens. Some kind of "this is you, this is them". I personally bought some cheap minis online and use monster tokens for monsters (either buy them or have a little circle that says MONSTER on it)
Recommended:
Players Handbook 1, 2, and the DM Guide. I would especially recommend this if you don't plan on subscribing to D&D Insider on Wizards website. The DM screen is something that I also highly recommend, to cover up what you roll, the stats on the monsters, and because it has a lot of useful info on it. I already own most of these things, but the DM kit seems like it would be a great buy for a new DM.