Many state have dot cameras 511 HOME
here is a DOT link for Nevada and as of now I do not see any chain requirements in the entire state. These are handy references for checking your route before you head out.
We have done the trip many times. The Siskiyous have great snow clearing equipment and you can use Oregon DOT cameras to see what is going on. TripCheck - Road Cams, Road & Weather Conditions in Oregon - ODOT
We check this regularly to get a feel for conditions. This is a dry year so I think snow may not be a big issue.
We call the DOT when we hit Medford and get the pass report and if it is good we go. Timing is everything watch the weather and pick your date accordingly. Doing so over the past ten years has made it easy for us. It's a haul but weather has never been the issue, it is just the mountains so it is a long day. Same with Shasta area.
We go from Coburg to Redding in one sitting in our MH and it is over. Once you get to Redding it is downhill all the way to the Grapevine basically. I-5 is kind of rough south of Sacramento but they have been doing a lot of repair work, you will be doing 55 all the way that is the speed limit towing. I do it in our MH, just relax you will get there.
The Grapevine is a big hill 7% and long then it drops you down into Northern LA area. I have done it a lot and there is nothing to recommend it except it gets you where you are going. I cut over on the I-210 and stay on it until the I-10 which will wind you up in Phoenix. It is a days worth but again avoid peak traffic times and things move along.
I have a reason to go to Paso Robles so I cut over to the 101 around Santa Nella Villages. There is a park in Santa Nella, a one nighter. We have found it handy to stay there because there is not much around that area and it is convenient. Not a great park but they have wifi and had some new managers last year who were working hard on the place. From the 101 I get over to the I-210 as fast as I can Hwy 101 into LA is not fun at all.
Depending on your route through Nevada you can get into some elevations. If you are on 395 in Northern California the elevations are high and there is kind of a dearth of facilities above Reno. Nevada highways are really nice but I do not know how they are managed for snow since they are state highways and not interstates. They are really fine roads however and used by truckers and in good weather a far better more scenic ride than down the I-5.
You plan through Bakersfield is as good as any and for your first time maybe the Interstates are better. But you could plan your return trip a different way up through Nevada just a little later in the year. It is beautiful.