You have to be registered as Republican or Democrat to vote in the Utah primaries on Feb. 7, and if you want to register to vote or change your party affiliation, the form has to be postmarked by tomorrow (Monday, Jan. 7). Here is the link to download the Utah registration form (to register to vote in Utah, you just have to have lived in Utah for 30 days before the election): http://elections.utah.gov/VoterRegistrationForm.pdf
You can fill it out and mail it to the county clerk's office--for Salt Lake County it is:
Salt Lake County Clerk
2001 S. State Street #S1100
Salt Lake City, UT 84190-1051
I think you can also register in person at the county clerk's office up until Jan. 21.