Have a look at the Help in the picaxe menu. Read through all the 3 files (you can skim read - this is just to get a feel for what the picaxe can do). Then try flashing a led. Then sample a pot and turn a led on when it gets to a certain value. Try sending the pot value back to the PC. Turn on a relay. Measure something like temperature.
If you can do all these things, then projects will appear out of nowhere! Eg, my car engine kept overheating. So I strapped a LM35 onto the radiator hose, send this into a 08M picaxe and turned on the relay for the radiator fan with a transistor relay circuit (as per the Help). A simple one hour project, but the first step is to play with the chips.
What sort of projects do you have in mind?