The first thing I do when I set off is to check that nothing rattles, all the nuts are on and tight.
This typically consists of checking:

  • Pedals
    Check pedals don't rattle/move where they thread into the cranks (obviously, they should rotate). If not, it is pretty certain that your seat is the wrong way around, as pedaling forward (normally) tightens them due to the way the threads are wound.
    Tool: 15mm spanner.

  • Cranks
    The cranks should not rattle/move (obviously they should rotate) about the hub/axle. If they do, tighten them (but not excessively, you will destroy the crank).
    Tool: Crank puller/extractor/tightener.

  • Bearing housing
    Check that the nuts holding the bearing housing are tight (assuming you have split bearing housing). If they can be undone with fingers then they are not tight enough. If this is the case, tighten them as much as possible with your fingers, then tighten a 1/2 or 3/4 turn with a spanner.
    Check that the wheel spins freely and doesn't rock backwards when it stops. If it stops too quickly, loosen the bearing housing. If it rocks backwards when it stops, the bearings may need replacing. However, rocking backwards may be caused by an inbalance in the wheel, in which case the bearings are good, look for age and grinding as other signs.
    N.B. Tightening bearing housing too much will destroy the bearings and/or make the wheel grind/resist when you pedal.
    Tool: 8/10/11mm spanner.

  • Tyre pressure
    Get on the unicycle, or put your weight on it some how. From above the tyre shouldn't look like it is squashing outwards, i.e. the squash should be barely noticeable. If not pump it up. A unicycle tyre needs to be pumped up more than bicycle tyres since all of your weight is on one wheel, not two.
    Tool: Car/cycle pump.

  • Seat nuts
    Check that the nuts holding the seat onto the frame are tight (not un-do-able by fingers). If not, tighten them until they are. If the seat is knocked/dropped often these eventually get loose.
    Tool: 8/10/11mm spanner.

  • Bumper screws
    Check that the screws holding the bumper onto the seat are tight (get your fingers under the bumper and pull gently). If not, tighten them, as these get loose when the unicycle is dropped. Your bumpers stop your seat getting torn to peices, so this is more of an aesthetic consideration.
    Tool: Cross-head screwdriver.

Things that need checking less often include:
  • Tyre wear
    Check that the tyre's tread isn't worn out. Obviously, if the tread has been worn out the tyre won't grip the ground, and riding will be difficult. Unfortunately, the only remedy is to buy a new tyre.

  • Tyre hotspots
    The tyre will wear more in certain places, due to idling and turning on the same spot. These are called hotspots. To prevent them affecting your ride by turning or idling on bald tyres, deflate the inner tube and twist the wheel about 90° every so often.
As well as these checks, you may have to deal with the following:
  • Creaking cranks
    This is a symptom of loose cranks, the creak comes on the downward push on the loose crank's side. Tighten the crank and nut with a crank puller. If the nut is not moving, tap the crank in (gently, don't damage it !), then try again.
    Creaking cranks can also be caused by grease on the spline (the tapered bit), remove the grease.

  • Creaking spokes
    After a while, the spokes may slacken. This weakens the wheel making it more likely to buckle when put under pressure. Tightening a wheel should be handled by an expert (just go to a bike shop), but you can approach this if you know how.
    Tool: Spoke key.

Cranks

I thought I should explain cranks. What they are, how they are attached, and how to change/loosen/tighten them, as this was not obvious to me when I first started out.

Cranks are the bent peices of metal that connect the pedals to the hub/axle of the unicycle. The axle has a tapered, square peg. The "fat end" is on the axle, and the "thin end" is threaded. The crank has a square hole in the "hub end", that fits the square peg on the axle. A nut fits onto the threaded end, on top of the crank. Tightening this nut forces the crank harder onto the tapered peg.

The fit between the crank and tapered peg makes it so that it is very hard to remove with pressure from your feet or arms, even if the nut is off. This is where the crank puller comes in.

One end of the crank puller has a socket shaped to remove the nut from the tapered end of the peg.
For the cranks to be removable, inside the hole in the crank surrounding the nut is threaded. The other end of the crank extractor screws into this. To force the crank off, the crank puller has a peg which, when tightened, will press against the axle's peg and pull the crank off.

To tighten the crank, either tighten the nut, or tap the crank in gently then tighten the nut.
To remove/loosen the crank, loosen the nut, then pull out the crank as much as you want with the crank puller. Most crank pullers have a way to attach a spanner to tighten the "pushing peg", as the crank is pretty (and so it should be) tough to get off.

Kit

As an example, here is what I bring with me on a typical unicycle ride:


Sources:
rec.sport.unicycling
http://www.unicycle.uk.com/maintain.asp
ATRer
Me


Stupot: "Also, if the wheel rocks backwards after stopping, it may just mean a slight imbalance on the wheel or tyre - this will only occur if the bearings are very good! (ie. they don't need replacing)". Also corrections and suggestions.
Albert Herring: "And creakign cotterless cranks are generally due to grease on the spline (which should be bone dry) rather than looseness per se."
Stupot & Albert Herring: Seat can be put wrong way around, not pedals. Pedals cannot be put on wrong cranks (I tested this).