I finished playing the D&D beginners Red Box with my friends. Now how do I start DMing another game? Where do I go from here?
I highly recommend adventure that comes in the official Dungeon Master's Kit: The Reavers of Harkenwold.
It contains many of the standard tropes of this genre of role-playing without being trivial or trite. My 24yr old daughter was a first-time DM and ran this adventure quite successfully.
The kit is a box that includes the poster maps, the adventure as two "modules" and all the paper tokens for the characters, NPCs, and monsters in the adventure.
It's a great way to bootstrap your D&D 4e campaign. There is an unusual amount of internet support for these adventures - with adventure logs posted at places like Obsidian Portal, missing maps created by users, and even 3D terrain (if you're so inspired).
There's really three ways to go here, and it depends on the focus of your game:
The Epic Game For the GM With No Ideas
The Epic Game For the GM With No Ideas follows pre-written scenarios, and is of all of them my least favorite, because I have issues with scenarios and my groups. However, if your groups are happy doing prewritten scenarios (mine go off to kill insert beast here instead) then you can look up a D&D campaign line. If you have a DDI subscription there's a massive archive on WOTC's site (here), but I've never used my DDI subscription for that, though you can also find others on various sites, for free or with a subscription or per-episode fee.
The nice thing about a scenario-based game is that it strongly cuts down on your need for personal involvement, nice for busy and/or unfamiliar GM's. My side note here would be that if you're not sure whether or not you're busy, you are.
Do It Yourself Epic Storyline
Writing a campaign from scratch is difficult, but if you really want to you can create an epic campaign simply by looking at a source for inspiration (I like to take a somewhat obscure story and then just rename all the characters and let the players run with it, instead of whoever the protagonist would normally be, but depending on how savvy your group is they may not enjoy this, or they may spoil things for themselves).
The main disadvantage of this is that you have to write it; most professionally written campaigns or scenarios have tips for when things go off-the-rails, and if you're a novice you might not. D&D's a relatively good system for recovering from stuff like this, though, even for a relatively green GM.
Patchwork is basically when you eschew a campaign-scale narrative in favor of individual scenarios. What I did in my first game was run the basic introductory adventure (this was Shadowrun 3rd Edition, so it was a not particularly great starter in my opinion), then played the entire game as action/reaction segments, letting the players choose what to do. This can be hard for a novice GM (like the time when I did my math wrong and my players made off with $2.5M or so of gold bars), and some players will be put off a little by the lack of a defined progression.
One nice thing about a patchwork game is that if you find an adventure that's not part of a larger campaign floating about on the internet or on a dusty shelf you can just plug it straight in, since it's sort of an "Adventure of the Week" game anyway, and if you decide to make the game suddenly go into an epic plot you can easily do so with some of the characters your players will meet along the way.
This is not exactly tabletop-related, but there's an open source game called GearHead (a roguelike about giant anime mecha), that has a random plot generator that basically does what you can do with the patchwork campaign, albeit at a more crude level. It even takes important NPC's and hooks them into the plot for the more epic segments. If you take the time and dig, it has some really good archetypes (it's primarily made by an English teacher/cartoonist, so the individual plot ministories work pretty well). You can find it here, and all the files for the storyline generation will be contained in a "Series" folder wherever the game's executable is. Disclaimer: I worked on a GH creator-sanctioned tabletop game adaptation for a while, though it's on hiatus until I refine my design skills.