If you absolutely *must* make a laser completely from scratch in order to satisfy your ego, I would suggest a ruby laser.
The power supply is very easy, and the laser head itself is quite straighforward. Sure, the ruby rod is going to be expensive, but that's the only part that will be really pricey. The rest of the components can probably be found in a surplus shop or on e-bay. (For the record - stay away from unfinished ruby boules. The optical precision needed to turn a raw boule into a finished rod that will lase is beyond the reach of all but the most experienced people.)
Yeah, a ruby is pulsed-only, but it *is* visible. If you build it right, you should be able to pop balloons with it and burn holes in things. Not a bad science project...
Forget about ion lasers - either argon or krypton. Bad idea. Way too hard, for the reasons mentioned above and a hundred other ones. HeNe will be equally difficult, but for different reasons.
A CO2 laser would be a nice challenge, but it's certainly doable - even with nothing more than a high school science lab to work with. Still, you're back to an invisible beam again. The N2 laser is probably the easiest, but again - invisible beam. :-/
A ruby laser is a nice compromise though. Go to the library and check out "The Story of the Laser" by John M Carroll. In the back (around page 230 or so) there is a description of how to build a ruby laser that should get you started. Show it to your teacher and see what he/she thinks.
Remember too, that even a ruby laser will have deadly voltages on the PSU capacitor bank. Any laser construction project is going to carry risks. But if you're careful, I think you can expect to complete a ruby laser sucessfully.
Adam