We recently moved from FL which had the homestead law which provides the right to use deadly force upon anyone I felt threatened of in my home or car - whether I had originally invited them in or not.
Welcome! I second the suggestion to look at the other thread, and to get a copy of Vilos' excellent book, but in brief:
Utah's law isn't quite as good as Florida's in this respect. Your car is not an extension of your home, so the "street" laws apply there. In your home, you do have extra protection, but it's not going to apply if you invited the bad guy in. Basically, if someone breaks into you home and you shoot them, the law says that you are presumed for both criminal and civil purposes to have acted reasonably.
On the street, you have no duty to retreat in Utah, and you can shoot in defense of yourself or others, or to prevent the commission of a "forcible felony". The list of "forcible felonies" is defined in state law, but it generally includes any felony that involves force or the threat of force. So, robbery, burglary, rape, etc. all qualify. Also, you don't have to be the target of the felony, but you do have to witness it, and you can only shoot to prevent it. Can't shoot the robber as he's fleeing.
On drawing your gun -- don't draw unless you're going to shoot. Drawing a gun on someone is assault with a deadly weapon (even if you don't shoot -- drawing is a threat and a threat of assault IS assault in Utah), so you need to be justified in using deadly force before you draw.
One other issue that may be different (I'm not sure) is that deadly force is not generally justified in defense of property. You can't shoot someone for stealing your car unless you or someone else is in it.
On carry locations, in Utah you can carry anywhere except a federal facility, a secure facility or a church or private residence whose owner prohibits it. You don't need to ask permission of the church or residence owner, it's on them to notify you of the prohibition either via a state web site (only the LDS church is listed there) or by posting signs or informing you directly. With a CCW (from any state)
you can carry in schools, including universities. Utah has full pre-emption, so no city, state or county entity can impose stricter regulations.
Hope this helps. Note that IANAL, but I'm pretty sure this is accurate. I'm sure if I made any mistakes, I'll be corrected