Pretty much all motor drivers these days use MOSFET transistors.
Because your motor is unipolar, you need four transistors. Four N-channel MOSFETs will work just fine, and you can buy power MOSFETs with logic-level I/O for about $1 each from places like Digi-Key. Adafruit also sells them.
You could also get a dedicated unipolar motor driver circuit, but I don't think you'd get anything out of that that you wont' get out of just discrete MOSFETs.