[RPG] What do I need to keep running D&D after finishing the Red Box

You're new to RPGs and you're looking for the sign that says "New GM Orientation"… but there isn't one. Most of us learn by doing, or by playing with a more experienced GM. There isn't really any beginner's bible. We do have some already-answered questions around here that should give you some context for your experiments, though.

1. Consider if you actually want to GM right away or if you'd rather learn by playing in someone else's game. (Or if you're open to both.) That will affect how you approach the rest of the process.

2. Find a group. You need one of those either way. It can be online or in person, but it's easier to learn the craft of GMing in person, whether learning by doing or learning by playing in someone else's game. You might have to explain what roleplaying is if you're pitching it as a new activity to friends, so reading up on that might be good. There are some good examples of what roleplaying in D&D is like as well.

3. If you're GMing, figure out how to start the actual game. There are a few different ways you can start the game. Eventually you'll learn that how much and what you want to prepare before the first adventure is just a matter of your taste and GMing style, but until then you have to just get started and see how it goes.

This is just a very high-level overview of the process. Follow those links to the detailed sub-steps of each of those, with their own variety of decisions and possible different paths you can take.

From there it's all a matter of growing in skill and techniques through experience. GMing a game has a lot of bits and pieces to it – plot, developing situations, improvisation, acting, crunching math for NPC stats – and it can feel overwhelming. Greg Stolze has written a good walkthrough of the things that a GM does, putting them in context, giving advice, and presenting them in a not-overwhelming way and order, so that the bits all sort of fit together. It's called How to Run Roleplaying Games, and I highly recommend it. It's not the only way to understand GMing, but it's a good summary of the process of GMing in a mere 13 pages, and gives some things to try. It's worth a look.

The system is designed to accomodate this

...but without the DMG it's a little tricky.

The science:

Basically, each enemy has an XP value. This is how much XP it's worth when it's defeated (divided among those who defeat it), but it's also useful for building encounters.

Here's how you build an encounter in 4e: You take the XP value of a "standard"-type enemy of the same level as the party, and multiply that number by the number of PCs in the party. The result gives you a "budget" that you use to "buy" enemies to create an encounter of average difficulty (the party is unlikely to die, but will expend a noticeable amount of resources --consumables, healing surges, daily powers-- during the fight).

For a more difficult fight, increase the level of the standard-type enemy whose XP you're using as the baseline multiplier to get your budget, up to four or five levels above the party. For an easier fight, drop the level down by three or four. The extreme ends of this will produce boss-level fights, or make-the-players-feel-invincible routs.

The art:

The DMG recommends actually using enemies up to five levels higher than the party for boss fights, but in my experience this is more frustrating than interesting; it's better to use "solo" type monsters of the party's level. The challenge level will be similar but more fun.

Combine standards, elites, and minions for interesting fights. Minions die quicker and elites last longer, so if there's an NPC or ability you want to be present throughout the fight make it a tougher monster type.

Use soldier (defender) and lurker types for longer more drawn-out battles, brutes and strikers for shorter, more intense fights.

If you've got a combination of enemies whose abilities support each other in significant ways, or you're adding strange terrain, remember that this may make the fight harder than its XP budget will imply.

The mechanics of monsters changed partway through 4e's tenure

With the publication of the Monster Manual 3, monsters got their hp reduced, their damage increased, and their powers were made a bit more interesting. This makes fights take a little less time while being a little more tense, but the overall resource drain per fight is pretty much the same. If you can get your hands on them, use post-MM3 monsters whenever possible until you're familiar enough with them to adjust the earlier monsters to fit that ethos. If you can't, don't worry about it too much. You'll learn to fiddle with monsters based on experience, and until then the fights will be a little more tedious than they'd be with MM3 monsters.

To that end, seriously consider a D&D Insider subscription. It provides a searchable compendium of every mechanic --rule, monster, item, class, race, etc-- ever officially published, a solid character builder AND a solid monster builder, and downloadable access to all the Dungeon and Dragon magazines for 4e. AND all the errata are kept up-to-date across the compendium and builders. I was suspicious of the service at first, but quickly found it to be nearly indispensable.