By applying an electrical voltage to water, you can generate hydrogen at one electrode and oxygen at the other. You can set things up so that the hydrogen is kept separate from the oxygen. If you make sure that no spark or flame comes near the hydrogen it won't explode. You can breathe the oxygen, but I would allow it to mix with some air as I'm not sure if it is very healthy to breathe pure oxygen.

There's tons about this on the Internet. Do a Google search of "school electrolysis experiments" or something like that. You will find literally thousands of them. I've put links to a few at the end of this answer. You should search on your own. Electrolysis of water is also in almost every introductory chemistry book, and there are probably several in your school library.

Simple experimental setup to generate H2 and O2 from water.

Audio tape of Glenn Shaw describing childhood electrolysis experiments.

Apparatus and setup for electrolysis of water