The California segments of Route 66 have been largely replaced by various interstates or surface streets. You can still drive from Santa Monica out to Victorville on various surface streets that were once Route 66, but the magic is gone in the LA basin. And, I wouldn't recommend driving the entire 80 odd mile west-east LA basin on surface streets towing a trailer. You can pick up a small segment around LaVerne, over to San Bernardino. After that, I-15 essentially occupies that space out to the Victorville/Barstow area where you can pick up I-40 over to Needles. There are various segments off the I-40 between Barstow and Needles, but the route is essentially the same. At Needles, the old Route 66 went up and over Oatman, not necessarily an easy tow with your trailer. I-40 goes around that mountain heading toward Kingman. At Kingman, you can pick up a fairly long stretch of the old Route 66 that rejoins I-40 just west of Williams. It's an amusing drive, complete with Burma Shave. Be sure to stop at Seligman, AZ, a veritable monument to Route 66 with lots of old gas stations, classic motels, and
1950 kitsch (including velvet Elvis). Further east, east of Flagstaff, there are small segments through Winslow, Holbrook, etc. I enjoyed driving through Winslow, where you can get your picture taken "Standin' On The Corner". On the east side of Winslow, there's dozens of old, classic motels/motor inns, long ago gone to seed. In Holbrook, there's a wigwam motel, compete with several old classic cars.
Beyond AZ, I can't be much help. I've traveled across NM, but don't recall any particular Route 66 segments.