Rout assistance Vancouver, BC to Arizona

You would be better off wintering in southern CA - Arizona is getting WAY OVER-CROWDED with Snowbirds.

Before I retired I used to take the month of January off and head south for that time period. I would take I5 through WA, OR, and CA to Bakersfield. At Bakersfield I take 58 east, get on I40 and then decide where in AZ I wanted to go.
I only had to wait for the interstate to be cleared in Oregon pass one time in 10 years. I always crossed the passes on I5 in the middle of the day when temps were good and roads had been cleared. Nine times out of 10 the pavement was just bare and wet.

If you are not in a big rush, another route that could work for you is Poco to Kamloops via Coquihalla, BC5 and Yellowhead to Edmonton south bypass, Alberta 2 to Ft Macleod, Alberta 4 to Coutts, I15 to Vegas, and on from there...... Lots of passes, but easier to handle IMO.

EDIT:
Easier to handle in good weather! Some of BC and all of Alberta (AFAIK) do not use salt/brine or anything but sand as an ice control method because of the ineffectiveness of salt in any temperature below about -12C. Given the unpredictability of mountain weather at this time of year (or any time for that matter), I would stick to I5. I get the impression that you are not fond of mountain driving........ IMHO, the northern US states have far superior snow/ice control than does western Canada.

Your question didn't include where in Arizona, or when you would be taking the trip, so there is some guessing here.

The route we use is similar to AFHauler's.

I-5 and 99 to Bakersfield. 58 over Tehachapi Pass 3,771 ft (1,149 m) to Barstow and I-40. We will be using this route to return from our next big trip.

Other options - I-5 to LA. That takes you over Tejon Pass 4,144 ft (1,263 m) and then through greater LA traffic. Then East on I-10.

Already mentioned is Siskiyou Summit on I-5. It's the highest pass on I-5 at 4,310 ft (1,314 m). It also has steep and long grades on both sides.

I-5 to I-90 to I-82 to I-84. Then you need to decide if you want to go the Las Vegas route or the SLC route. This route has some mountains and snowy passes in the winter. We will be taking the SLC route Southbound for our next big trip.

Sally B wrote:
First time travelling from Vancouver area BC Canada to Arizona and am wondering what the best route is to avoid the mountains during our trip? Thanks for your route suggestions as we are new to this!

We usually travel the I-5 til Longview, Washington, then head over to the coast and travel down the 101 until around Santa Rosa, then cut across to the I-5, travel down to Bakersfield and head east on 58. We like to stay at the Orange Grove RV Park outside Bakersfield as it is unique and a good stopping place.

Our route is slower but beautiful and we mostly avoid the traffic of the I-5 and weather in the mountains.

I-5 all the way south is a safe trip, except you'll need to go over the Grapevine just north of L.A. if you take I-5.

In an area surrounded by mountains, it's impossible to travel almost anywhere without dealing with them. If you are going any time soon (Snowbird?), you might want to take MDKMDK's advice and stick to I5 to travel along the mountains rather than cross them as you would if you went east from Poco, then south. Fewer passes and a bit warmer weather. Grant's Pass and Siskiyou might be the toughest to deal with in a winter storm. I10 might be the easiest route to go east from LA, depending on where you want to end up.

I would pick a good weather window, and just do it.

Gotta expect snow in the mountains.

Traffic versus mountains.
You choose.

I-5 south through WA, OR, CA to the AZ border then east on I-8 or I-10.