Looks like this is a year old, but it's still a relevant question that I've been asked recently - so here's my two cents:
Permits can be pretty easy to get for interior renovations if you aren't replacing or changing the exterior of a building in San Francisco, so long as you aren't in a historic home. If you're renovations are relatively simple (ie: you're not adding on to the house or making any major exterior changes) you'll like get what is called an over the counter permit, which means you will submit plans to the city and get the permits the same day. I would agree with the commenters below that having an architect or contractor go through this process with you is advisable, as they are familiar with the process and will likely make it smoother for you.
Regarding what requires a permit: nearly everything does. Anything involving electrical or plumbing work will require one, and any structural changes (ie: moving walls or knocking one down) would as well. Switching out vanities in bathrooms or cabinets in kitchens would too. Things like installing new appliances, changing out fixtures, painting and the like probably wont need permits though.
You're best bet is to talk to a contractor or architect about what work you want to do. I work with an LLC that renovates homes in the city and have a fair amount of experience with this process, and I'd be happy to recommend a great architect or contractor, or help in any other way I can, with renovations.
Hope this helps!