Q: How do I design a game world?
A: Imagination, Pencil, Paper, Time.
You have to decide on some fundamental aspects (local and global), such as technology level, magic level, politics and Government, climate, species, gods, and Darwin! (development of life), ecosystem, etc.
Once you have drawn a map, and put in a few towns, roads, rivers, ponds, seas, forests, woods, cities, borders, countries, continents, races, cave systems, volcanoes, mountains, hills, underground caverns, trade routes, currencies, etc, you have to NAME them.
Each city or town will have a name, notable personalities, pubs, streets, entertainment, history, income, army, militia, police?, rules (real and pretend), guilds, thieves, water source, allies, enemies, language, population, population breakdown, races, racial-allies, racial-enemies, etc.
Putting you off? This is a lot of work, but remember only the places that the PCs will be affected by need a lot of detail, but the more information you have the better since it can potentially give you adventure ideas.
Also, try out http://hops.wharton.upenn.edu/~loren/rolegame.html for info on world building (I haven't checked this myself though).
Q: How do I go about designing an adventure?
A: Start with an idea, write down everything you think of. Remember a few rules.
Motive. The PC's must have an incentive to actually adventure. This can be anything from simple greed, to something based on a characters past (always have players write down their characters history).
Cause and effect. People don't always do stuff because they want to, they do it because they have to. The king raises taxes, the people can't afford to pay, thus is born Robin Hood. Victims of circumstance. Civil unrest, civil peace, drought, good farming years, etc will all effect people both directly and indirectly.
Chaos. People don't always do things because they have to, they do it for "the sheer hell of it". Random events can and do happen, but are usually put in as red-herrings to the main plot.
Encounters. What and who are they going to meet. Why are those people there, how do they eat, who do they work for. Don't just use the wandering monster chart because that does not explain why creatures are there. If you have time make your own for each area, for a random encounter. Better still, cheat, plan everything! It's always better as a player to know there is a reason for a monster to be where it is.
Rewards. What sort of character rewards are there? Land, money, magic, training, etc.
Q: What makes a good adventure?
A: PLOT, Surprise, Mystery, Romance, Puzzles, Magic, Horror, Suspense, and characters with personalities.
Q: What makes a good campaign?
A: A good plot-line, and series of sub plots. They don't have to be obvious to the players, but they should exist. Make sure that events not involving the PC's take place. A war might end, two nations might join, a river might flood, armies might be gathering, etc.
Q: My players want to design spells / I want to design spells, what do I do?
A: Write down the effects of the spell, the damage, area of effect etc. Usually the casting time is equal to the level, but there are exceptions. Compare the spell to existing ones and you assign a level, if you are not sure err in the side of making it more difficult (It not being a `common' spell means it is less practised, unusual, etc). Choose which sphere or school with regard to other spells. Don't allow spells which do exactly the same thing, at exactly the same level as a spell of a different sphere or school, because this defeats the object of the whole thing.
In the case of a priest spell, the character will simply pray for it (in most cases everyone who worships that deity will have access to the spell, sometimes everyone who can cast that sphere). The deity may, or may not allow for it, or may require special worship etc before the priest may use the spell. To show greater commitment the deity might have the prayer put into an exact wording, may require fasting, a special task, etc. The prayer may even have limited access within the same preisthood, causing possible political and personal tensions.
In the case of a wizard spell, the wizard must research the spell in a laboratory for a decent length of time. Further details are in the complete wizards handbook.
Q: I want to invent a new creature, how do I do it?
A: Before you get down the game statistics you have to ask the following questions: what does it look like, what does it eat, how long does it sleep, how does it breed, is it docile or violent, when, how was it evolved, what noises does it make, where does it live, what climate does it like, racial variants, etc. Not all of these are necessary, but do help to flesh out the creature.
Now look at it's game statistics, first compare it to natural creatures like horses and lions. Then find a creature that it most resembles (if there is one) and look at the game statistics for that. Compare any natural weaponry it may have with real weapons (e.g. are its claws the size of a dagger, or are they like a club?). Compare supernatural attacks with similar creatures, likewise with supernatural defences. Once you have the creature designed, ask a few questions: how would this creature effect local wildlife, what other creatures would hunt it, what creatures are scared of it, how does this creature effect the local and global ecosystems, and even what does it taste like!
Try attacking it with varying levels of NPC's (if you have the PC's character sheets even better). Try intelligent attacks and random attacks. In addition, try attacking this creature with other monsters varying from a chicken, a lion, a wraith, a basalisk, a giant, a werewolf, an owlbear, and a great wyrm. With each creature ask yourself which you would want to win and why, and adjust your creature accordingly. If you know someone who role-plays, but is not in your group, try and playtest it with him. Try it with the characters anyway, gradually fine tuning it. Put down any differences to racial variance and sub-species.
Q: I want to invent a new race, how do I do it?
A: Again, before you start with game stats, you must come up with their ecology, asking many of the same questions as above as well as: Is there a racial stereotype, how often is this deviated from, their psychology, their philosophy, their technology level, their ethos, their religion, their rituals and traditions, how do they relate to each other and other creatures or races, where do the live. With all of these question you also have to ask why.
Then you have to design a `typical creature' as for humans, elves, giants in the DMG (i.e. a creature of no level). Like with a new creature, you should compare this creature to others, and see what role it fills in the echo system (could the take over the world, would the be wiped out, etc).
Then see what `classes' the race can be, there is a guide chart in the DMG (and also in the monstrous compendium). This may depend on whether the race is humanoid or not. Is it a PC race or NPC only? How common are characters which have a character class?
Q: I want to invent a new magic item, how do I do it?
A: Like everything else the item should have a history to allow it to `really exist', the players need never know this but it helps give even an item a `personality': Who made it? Why did they make it? Did anything go wrong? Who was it for? Is it unique or was it copied?
Do you want to make the item a bauble, minor-magic, major-magic or an artifact? Again compare it with other items. Then assign the game mechanics to it.