Driving from California to Dallas, TX- what cities to avoid?

I have one recommendation:  The hotels between El Paso and Midland (Texas) are very pricey!    The hotel availability is limited and the oil field workers pretty much have them filled to capacity.  When we drive to Phoenix from Fort Worth, we overnight in El Paso or Las Cruces. There aren't many towns, so plan your trip to skip Pecos (Texas), Van Horn and Monahans for overnights.  
We also had a heck of a time finding our way back to I-20 after getting off for a food break in Pecos.   Between El Paso and Odessa-Midland there is just not much to write home about.  However the @Permian Basin Petroleum Museum is very interesting (it is in Midland) 
I can't speak about west of Phoenix, but both Phoenix and Tucson are great cities.  Rush hour in Phoenix is a pain but Tucson's interstate is on the edge of the city so you can just zoom right along.     You really won't have any place to avoid, other than trying to find a room in Pecos or Van Horn.  
Phoenix to the east side of El Paso is about 7 hours for us, but we are not driving with a baby.  El Paso to Fort Worth is 10 hours.    
We have not driven the North route through Flagstaff, Albequerque, and into the Panhandle.  
You will have lovely weather across Arizona and New Mexico if you are taking the southern route; pleasant balmy days and cooler nights.  It will be colder in the DFW area - it's 38 right now on Feb 9.  There is an old joke that if you don't like Texas weather just wait 10 minutes and it will change.  Bring things that you can layer and check the 10 day forecast for Dallas before you leave home.   If it is the end of March when you come we may all be decked out in our shorts!  Then sweaters the next day ....
There should still  be plenty of mountain snow across northern Arizona and New Mexico if you take that route.  Flagstaff is a popular area for skiing and snowboarding 
A fun entertainment break on I-20 west of Odessa is Monahans Sandhills State Park  It is a big collection of sand dunes, and you can rent a saucer sled and slide down the dunes.   Another nice spot is White Sands National Monument in New Mexico. It is north of El Paso.    Carlsbad Caverns National Park is on the eastern border of New Mexico if that interests you.   
It looks like you may be in Texas for Bluebonnet season.  They usually start blooming in the last 2 weeks of March and continue for about a month to 6 weeks.  In the DFW area they pretty much peak around the 1st week in April   A bluebonnet tour is a great day trip out of Dallas. 

Pecos (Texas) has great Texas canolopes when in season. This whole west Texas drive will be very boring for a child except playing in the Monahans Sandhills State Park between Monahansand Odessa (Texas).