SF part -- if you're going on be in the actual city itself, save your money and take public transportation -- Bart from the airport (SFO) to downtown and probably muni around the city. Parking in the city when you can find it can be very expensive. But if you're going to drive and park in the San Francisco itself, as a public service to those of us who live here a couple of suggestions:
1. get an automatic
2. CURB YOUR WHEEL. if you can't remember which way the tire goes, just ask yourself if the car were to roll, which way do you want the car to roll? and remember you want the car to roll towards the curb. that's what i do. the parking people really do give parking tickets if you don't curb your wheel. and that would be a budget buster.
3. use the park break. no, it's not optional.
If you're starting in SF and doing your grocery shopping there, bring bags to put your stuff in. If you need a bag from any store they'll charge you $0.10 per bag. not a huge budget killer, but just so you know. The bag charge also is in place in LA County. So depending on where you are in LA, possible grocery bag charges.
Grand Canyon, I would suggest topping off the gas tank before you enter the park. The only gas station in the park is on the East side as you exit. However you might want to let your credit card company know about your travels, so your credit card doesn't have a hold put on it for too many gas station transactions in a day. (yep, that would me, be.)
Looking over your 20 day time frame, unless you're really interested in SF and Yosemite, consider dropping those if you're tight for time. If you end up doing the drive from LA to Vegas, I consider stopping at Barstow to pick up gas and hit the McDonald's there for a bathroom break. I've only done the SF to Vegas thing, so for me I know the gas stop was really important. But I wouldn't want to worry about needing gas in the desert. It may cost you a little bit more, but the peace of mind was worth it. Not a lot out there.
The hotels may have winter rates, so check those out. Sometimes you can score a free breakfast. See if your hotels have a frig or microwave. That would really help get that $75 food budget down. If none of them do, consider a hot water immersion heater -- oatmeal and soup. It would about a $10/$15 investment for the hot water immersion heater, but that combined with a box of instant oatmeal, and you would get several breakfasts out of that investment, not to mention the hot tea, coffee, and hot chocolate possibilities.
ITA with mary w and the SF free walking tours. They are quite good, and they pass an envelop around after for you to put $ in. I've done 2 of those at the Japanese tea garden, which does have free times.
One last money saving gas tip, if you end up driving back to CA from AZ, fill your gas tank before you leave AZ. On 10 going westbound, if I remember correctly there is a sign that reminds you. Yes, the gas is more expensive in CA.
ITA with FreedomFound -- Arches is beautiful. And if you're going to add Arches, consider staying in Moab and also going to Canyonlands. Arches and Canyonlands are very close to each other.
Sounds like a wonderful trip. Have a great time.
Fell free to PM me if you have any additional questions.