How much of a budget are you on?
I'm sure it can be done cheaper than I go, but I budget at least $50/day that includes gas, food, camping and the occasional treating myself to a good burger somewhere.
If you do 400 miles a day, that's about 2 tanks of gas which alone can be $30 (in the expensive parts of the country). If you're not back country camping, some campgrounds are $5, but most are closer to $20 and up. I usually shoot for the $10-$15. That's up to about $40/day. If you stop at grocery stores regularly, you can save some money on food, but less than $10/day on food would be hard for me. Then there are the little incidentals, the bottled water, soda, coffee, beer, snacks, etc that you might want to get. If you visit State or National Parks there are entrance fees too. It can add up pretty quick...suddenly your looking at a minimum of $50/day x 28 days = ~$1400 and that doesn't include gear if you need to purchase any...I don't want to burst your bubble, I just want you to have a realistic picture of the actual cost involved...
Maybe someone else will chime in with tips for reducing your costs. I guess it depends in part on how much you like to rough it.
The range in temps and climates can be a lot between Arizona and the Pacific Northwest.
FWIW, I had a below freezing morning in the Southern Cascades in Oregon in the end of July last year. I was not expecting it and was very minimally prepared. Thankfully the RV next to me had about a months worth of wood and they were very generous with it...
Keep us posted about your trip, we'd all love to know about it. If you need a place to stay in SF, let me know, depending on the in-laws' plans we may be able to put you up for a night.
2011 Imperial Red
Givi V46 / V35's SW Motech mounts
Skidmarx hugger Moto Werks bar riser
Mondo Motors LED Driving Lights
A little bit of road rash