By motorcycle[edit]
Rte 1095 isn't as bad as people make it out to be. There isn't much traffic and you can hear the cars and trucks coming. If you're a little adventurous, rent a motorcycle in Chiang Mai and make the ride up to Pai. You can stop at the waterfalls and small towns along the way, and you'll really enjoy the trip, as opposed to being motion sick in a bus for hours, and being forced to stop at the driver's friend's restaurant. Make sure to take some warmer clothing on your bike, as it tends to get a bit chilly in the higher portions of the ride. As a novice rider, expect the trip to take around 5 or 6 hours, including stops at sites and restaurants along the way.
aYa Service [1] offers one-way rentals from Chiang Mai to Pai (or vice versa) with luggage delivery for "free," though they charge extra for returning a bike in a new city after a 1-way trip. They will keep your passport and send it along with any baggage to aYa in Pai.