If driving over mountain passes in most Western states, you must have chains on board, even if the "Chains Required" sign is not displayed.
If the chains required sign is displayed, and you are towing ANYTHING, you must have chains on the drive axle of the tow vehicle, and IF the trailer has brakes, you must have chains on one axle of it. On some passes there will be a check station where they will inspect your vehicle, and turn you back if you don't comply with the requirements!
When driving with chains, do not exceed 35 MPH. Personally, when I have to use chains I don't drive any faster than 25 to 30 MPH. If a cross link breaks it can cause severe damage, and it is much more likely to break one at higher speeds!
I have lived in "snow country" most of my life, and if chains are required I will stay home if at all possible. I recommend you do the same. If chains are required, find a place to park and stay there until the roads are clear!
Even so, get M&S rated tires with an aggressive traction tread for the tow vehicle (front and rear) at least, and maybe even for the trailer!
Good traction is a must when driving on packed snow and ice!