Hello, former San Diego County person here. As you have noted, there are a number of ways to get where you want to go next. We used to tow a travel trailer as well, so I will try to give you the run down on where the mountain passes are that you are going to have to traverse.
Leaving Lakeside to get to JTNP: Plan A= north on CA-67 towards Ramona, then east on CA-78 through Julian (gorgeous). If you are traveling on a good day, a stop at Dudley's Bakery in Santa Ysabel is good for the palate (but terrible on the weight....). The route 78 between Julian and where it connects to CA-86 is scenic but desert. At this time of the year, though, you may catch some beautiful desert flowers in bloom! At CA-86, head towards Mecca/Coachella (probably signed west) which takes you up to I-10 while passing by the slowly dying Salton Sea. This avoids the major cities. Mountain passes okay, one will be slow but it's not excessively long.
Plan B) Very similar, except you head out on I-8 east to CA-79 north. That scenic route takes you through Cuyamaca Rancho State Park. It will drop you in Julian, where you pick up CA-78 east and continue as Plan A indicates. But you'll miss Dudley's Bakery (a local favorite). Passes okay, see Plan A.
Plan C) Get on I-8 west, take I-15 north to CA-60 east to I-10 east. Lots of city driving, but still faster than the first 2 if you're "in a hurry". Mountain passes are easy grades, as they are interstates.
With the snows and rains I've seen on the news, I gave you the three as alternatives. Check locally!
As far as Quartzsite is concerned: many parts of the desert around here are open to long-term "boondocking". It's just desert, and could be in bloom at this time of the year.
Plan D-includes Quartzsite: I-10 to Quartzsite. Head north on AZ-95 to Parker (more desert), then continue on AZ-95 to Lake Havasu City (riverfront then lake) and on to I-40 just east of Needles and Topock. I-40 east can take you to the GCNP. Plan D-2 includes taking AZ-66 from Kingman up to Peach Springs (inspiration for the movie CARS) and back to Seligman where you continue on I-40 east to AZ-64. Suggest you have reservations somewhere, at Williams or GCNP, for your trailer (goes for JTNP as well).
Plan E includes Quartzsite but more scenic approach to GCNP: I-10 east to Quartzsite and further east to the connection of US-60 east (exit 31). At Aguila, catch AZ-71 north which will connect with AZ-89 north into Prescott, then Chino Valley and up to I-40 at Ashfork. One mountain grade, if I remember correctly, not too bad.
Plan F includes Quartzsite with mostly interstates: I-10 east to Loop 303 north to I-17 north. You can either take 89 up to the east entrance of GCNP or take I-40 west to Williams and AZ-64 to the south entrance. All grades on I-17 are interstate mandated, but they're long.
Plan G - no Quartzsite - several ways to get up to I-40. Many would go from Desert Center using CA-177 to CA-62 east to US-95 north to Needles. If adventurous, and wanting a little more old US-66 feel, head to CA-62 and catch the road up to Amboy, and head east to Essex.
Hope this helps.
Donna