How do you start a new FRC team?

[Siri]

Honestly, the easiest (not that it's easy) way is probably to recruit a veteran FIRST mentor(s), preferably a teacher at the school. FRC is an amazing experience, but the learning curve can be pretty steep. If your friend is near your team, maybe start with a sort of joint setup. Regardless, s/he would benefit greatly from making friends with teams/FIRST in their area area. (I'm not in CA, but feel free to contact me as well. CD is also, as you've already realized, a great resource.) As to your specific questions:

1) What do we need to start a team
As others have said, you need students, mentors, space+tools, and money. You also need insurance. Parent and school support will make life a lot easier as well. In terms of people, best case is to get those interested (experience is nice too) in both the engineering and non-engineering sides.

Finding Space: The school/school district could be very helpful here. The students may already be there/nearby, the parents are used to it, and it can make insurance easier. Otherwise, talk to local technical companies--big and small--and local teams.

Insurance: If you can't go through the school (and do check what this covers), 4-H and other organizations can be helpful in this respect. Check out this thread for more information.

2) Where to look for sponsors
Apply for a NASA grant.
Other than that, the companies your students' parents and mentors work for are usually a good place to start. (Of course, this requires recruiting students and mentoring simultaneously.) Follow those leads: a lot of your local STEM professionals/mentors will know possible contacts. Other than that, I took to typing "hardware", "technical", and "office"...and "computer", and "metal" (you get the idea) in Google Maps' Search Nearby feature.

3) How to go about getting sponsorships from companies
Best case, make a sub-team for it headed by a dedicated (preferably experienced/knowledgeable) mentor. Get the students involved: a lot of successful teams have quotas for their students. 1511 has an especially good and well-documented method. 1511's Patron Drive.
As for what exactly to present, as a new team this may be difficult. You won't have much personal history to go on, but FIRST in general has some very impressive success stories and statistics that might be helpful. If you can recruit experienced mentors, etc or help from local teams, that would probably help with convincing as well. Make a budget (I suggest starting small) and lay out your plans, why they're important/beneficial, and why they're likely to succeed.
ChiefDelphi's Sponsorship 101
NEMO: Making a Team Packet
1640's Prospective Sponsor Handouts and Presentations

Also, don't limit yourself to just sponsor recruitment. Community fundraisers: car washes, pancake breakfasts (we partner with Applebee's), spaghetti dinners, pie sales, etc can actually be very profitable.
ChiefDelphi's Fundraising 101

4) How to Recruit team members
Recruiting students: Definitely get in the schools. Recruiting without school support is very difficult (we've tried). Talk to teachers. You can ask them to plug your team, point out students, even help out themselves. We do demos at Back to School Night and during state standardized testing for grades not testing (the PSSAs in Pennsylvania). We're also in the teacher's newsletter and the school's morning announcements. I definitely recommend getting as much support as possible, both from the the administration.
As for specific subjects, talking to everyone is great, but especially consider physics/science, tech ed/wood shop, computer science, CAD, other engineering courses, etc. Also, FIRST has a huge non-engineering side including logistics and organization, public relations, fundraising and grant writing, media and promotional materials, and several awards that revolve around business planning and community outreach activities. That means kids who like business and organization, writing, art, graphic design, etc.
The easiest way to get school support is by having parent support. Conveniently enough, you'll also need parent support anyway.

Recruiting mentors: Our mentors tend to find us: student's parents, local technical company employees, people who see us at community days and school events (even if they're kids aren't old enough for FRC), FIRST alumni. (Do watch the college student angle though. It's worked for others--it's worked for me thus far--but do be careful about how much time they can really commit.) Unfortunately, we don't have active teacher mentors, but I do recommend pursuing this angle. They can offer a valuable skill set and make school relations easier.
Overall, the earlier you start building your mentor base, the easier the process will be. Talk to local teams, the schools, etc. We've gotten a few mentors in the past from retiree volunteer organizations. Retired engineers, machinists, secretaries, etc might really help out.

5) How to Organize the team
Check out Team 341's Team in a Box - great team, great resource, wonderful people.

I'm not sure what level of organization you're thinking about, but in general, there are more organizational methods than there are teams in FIRST. Autocracy or democracy, school club or business model, mentor- or student-run* or both, they all happen on many different levels of success. Once you've jumped the more basic hurdles: ensuring everyone knows where and when meetings are, etc, the organization really depends on the people you've got and what you all want to do. Really to think about this (what you want and what you have), too. A lot of very successful teams have published informative handbooks (a good idea in general) that lay out their team structure. These would definitely be worth a read to investigate what you'd like to try. I've assembled links a bunch of them (here), if it's helpful.

Other than that, I would definitely be careful about not getting in over your head. Try to have one very dedicated person (usually a mentor), preferably not already otherwise thoroughly committed to the team, for every sub-team you start. These would include mechanical, electrical, and programming, but also things like fundraising, public relations, media, etc. Dedicated students are of course also a must, and having positive but not overly time-consuming ways for non-mentor parents to stay involved can also be a huge benefit.

*If you've got the dedicated students, it can be a great experience. (I know I learned an utterly unquantifiable amount as an active student captain-manager, but that interest seems rather rare on my team...) Very high levels of student leadership can be a good thing, but there's a significant argument to be made for the mentor-student interaction in FIRST. (Much of that debate can be found via the ChiefDelphi search engine, in fact.)

6) Anything that might help us start the team
Nothing else sticks out, but feel free to ask questions as they arise. Good luck and thanks for your interest in spreading FIRST!