Red tulips will give you purple ones eventually, just keep fertilizing them and be patient. You also get black tulips that way. I've grown both using the reds
Aside from purple and purple? It's the best way though! Other than that..for me, chances are higher by breeding orange+orange tulips--think I got my first purple tulips from this combo. (I think the orange should come from red+yellow for better results.)
Red+red is the most basic way but this could give you black, orange, purple, or even red. Not necessary a bad thing if you want those colors too. I just find it faster to breed hybrids from hybrids.