Laws for this are contained in RCW 9.41.300 and RCW 9.41.280.
Courthouses (storage required)
Places declared off limits to under 21 under the state liquor control board (bars, bars in restaurants)
Mental Health Facilities
Law enforcement facilities and jails (public areas of egress do not count).
K-12 schools and school grounds (with certain limited exceptions)
Assuming you have a Utah CFP, you can carry concealed or loaded in your car. Also, remember that in Washington, your pistol must be concealed from view when AWAY from your vehicle if you leave your pistol in your car. There is no requirement to conceal outside of that.