Isn't there a faster way to move from one channel to another on Sirius than one at a time?

 

You don't indicate a model year or country so I'm assuming your Sirius is US/OEM. You have 18 presets, 6 for each of the three Sirius banks. Read page 5-46 in the manual (2014 manual) to learn how to program presets.

Electronic US manuals are downloadable here:

MazdaUSA > MyMazda > Manuals & References