You can write for any instrument you like, but don't try writing harmony (two notes at once) - just melody, at least for now. You can use Sibelius or other music notation software, or if you like you can write it out by hand. If you run into trouble, please email me - I'm often hanging out near the CAMIL lab, and I'm happy to meet with you for help on composing. Good luck!
Here's a wikipedia article on twelve-tone technique; and here is a page (UIUC net id required) with many listening examples, some of which are twelve-tone; try listening to the following as an introduction:
- Arnold Schoenberg, Suite opus 25 for piano (7 short pieces)
- Anton Webern, Variations for piano opus 27
- Milton Babbitt, "All Set" (12-tone music for jazz band)