Yes! As long as they're legal for you to have both where you are (I assume that's a "yes") and where you're going (I gather that Utah is no problem either), then you're fine.
Once upon a time, there were instances of gun owners traveling across several states who had their guns seized by the authorities in states administered by anti-gun bigots. The Firearm Owners' Protection Act of 1986 cleared that up.
Yes, it's perfectly legal to transport your weapons with you. {EDIT: you've seen davis513's description of where they should be: e.g., unloaded in the trunk, or in a locked case in a van, SUV, etc.} I expect that a number of THR members have made trips from Texas to Utah with guns legally transported in the car. One of them might chime in to give you current details.
{Edit: I have slow fingers. Thanks, davis513}