How do I lay asphalt shingles on a roof, and what should be under them? Are they self-adhesive? Can I stick them directly onto plywood?
My other answer only answered "what should I put under the shingles". I reread the question and see that @Vebjorn asked how to install shingles. Doing a shed is a good project if you haven't shingled before. Doing a house is harder because of the peaks and valleys.
To shingle your shed, assuming a straight gable roof:
A. Roofing felt.
C. The peak
Asphalt shingles by themselves will keep out the rain. Extra steps increase longevity. You need to follow the manufacturer's directions (take photos!) if the warranty is important to you.
Careful preparation can double the installation time and increase the cost by 25%. Whether it's worth it to you depends on how catastrophic a leak would be. For me, my garage or shed gets minimal preparation since a leak doesn't matter that much - it can be patched when it stops raining. For a house or any finished space like a nice studio, a leak will mean ruined drywall, etc. so more attention is warranted.
In order from minimal work to a complete, nice job:
The shingles are not self-adhesive, but you'll see strips of tar on the backside. After you nail the shingles down, the sun will melt these strips and adhere each course of shingles to the one under it.
One thing about the double-course of shingles that you start the roof with. The reason you cut the tabs off and reverse the shingle is to get that strip of tar at the very base of the roof. The tar will melt and glue the first tabs down, helping prevent the wind from getting under the shingles. For some reason many people I know, including my father who I worship, will put an upside-down course of shingles first without cutting off the tabs, to save time I guess. To me this is pointless since it does not do its job, which is to lock down the tabs at the leading edge of the roof.
Depending on your climate you may need some additional material. My roof has an ice guard on it to keep ice from forming at the edges and backing up underneath the roof. I've also got roofing felt as a barrier between the plywood and shingles.
Roofing shingles are nailed down with galvanized roofing nails (at least mine are) and the little roofing work I've done has used a nailgun to nail the shingles through the roofing felt and plywood. The shingles have a compound on them (may be just plain old tar) that seals around the nail to prevent water from dripping through.
Unless they've come out with new self adhesive shingles you're going to have to nail each course to the roof.