Regarding your interest in eventually making your way West, San Francisco is a fantastic place to be queer/trans. I love it here; there are tons of resources, cool people, and all sorts of fun things to do, great art, music, culture, night life - it's a fabulous, beautiful city. However, it's also a tremendously expensive city to live in. The rents here are monstrous. Across the Bay in Oakland and Berkeley is more affordable, but that is still something you should take into account.
Try to find - or take from your house, if one's available - a large frame backpack, of the type used by backpackers, long-distance hikers and mountaineers. These are very spacious, much more forgiving on your back and shoulders than other types of bags, and will generally have spaces for a sleeping bag and bedroll.
Don't pack sentimentally - pack for survival. If you have important things that you can't take and don't want ruined, try to reason with your dad to store them somewhere safe. Taking a favorite book, your sketchbook or diary etc. can be a major emotional help, but when you're homeless, you need to avoid carrying too much stuff you don't absolutely need.
Unless you're absolutely sure you'll have a home wherever you're going to, pack camping gear. Even if you have a secured spot at some homeless youth house, it can't hurt to have a bedroll, sleeping bag and a small personal pop-up tent or a tarp and some rope. It's all a matter of how much room you have, and how much you can carry.
If you need to steal to survive, do it - but steal from large corporate chains, especially big-box stores. They're less likely to catch you, less likely to take action of they do notice you stealing, incur less loss, and generally suck anyway. Small mom-and-pop shops can't afford theft the way giant chains can.
Be selective when hitchhiking - if you get a bad feeling when someone pulls up, don't get in. Also be aware that homeless shelters, while valuable, are not always free of violence, theft or other danger.
Avoid drugs. This may sound patronizing or trite, but it's absolutely stunning how many homeless folks - queer and trans youth especially - end up hooked on some substance or other after being offered some. It becomes really, really tempting when you're homeless, miserable and cold.
Spend what money you have very, very sparingly.
Abandoned buildings can be tempting as places of shelter. However, do be aware that there are dangers in squatting, apart from just that of the police: structural weaknesses, exposed wiring, and of course the neighbors. If you decide to squat a building, make it as covert as possible: enter and exit via the back entrance or window, and be very quiet. Some areas have communities of punks who set up a communal home for themselves in a squatted building. These squats could be very worth seeking out, as they may be able to set you up with resources or even housing, and are much more likely to be trans-friendly than the homeless community at large.
Some of this stuff is pretty obvious, but it can't hurt to say.
And yeah, your mother's a lizard. Ech.
Best of luck! Make friends where you can, and travel safe!
