Now, it's been a long time since I saw a traditional QA team. They definitely still exist, but now that there is a lot more automated testing, things have changed quite a bit. However, I would still caution beginning programmers to stay away from QA roles. It can be actively damaging to your career as a programmer, unfortunately.
What should you do first starting out? Get an entry level job as a programmer. If at all possible, go to school with a co-op or intern programme. Try out a couple of places in your first 2 years and in the last 2 try to get into a place where you might want to work. Then stay there for a couple of years after you graduate. This is by far the most effective strategy I've seen for getting into the industry.
Otherwise, try to make contacts in other ways. The first job is super important, so you need to find someone who will give you a chance. Whether you have a traditional education for a programmer or not, you need that first job. I would recommend moving to a large centre (not necessarily SV) and going to every meetup that you can manage. Make contacts. Work on a side project. Discuss your side project with people that you meet at meetups. Possibly go to a bootcamp to meet other people just starting out. Make friends and talk about programming as much as you can. Eventually, someone will give you a chance.
The most important thing: take that chance. Stay at that job for at least a year (better 18 to 24 months). Then I recommend that you find another job and leave (in that order!). For programmers, it is usually best at the beginning of your career to work in a few different places. After you have 8-10 years of experience (3-5 places), then you will have a really good basis for building your career. Again, a co-op type education can short circuit a lot of this because you can work at 2 or 3 different places while you are studying -- it is really valuable IMHO.
If you are really struggling and can't find that first position, it might be an indication that this is not the career for you. I've seen lots of people offer their services for free just to get the opportunity to "break in". Don't do that! Every person I've seen do that has been horribly misused and they have never turned it into a career (again, my experience: I'm sure that it has happened somewhere, but I submit that it is very uncommon). There are lots of jobs for programmers. This is not acting. You don't have to eat ramen for 20 years before you get your big break. If it's not happening, there is probably a reason (as difficult as it is for me to say). If you are absolutely determined to make a go of it, then the best thing to do is bet on yourself and write some code (whether as a hobby or as a business venture). If you manage to make a successful free software project or business venture you will be sure that someone will hire you later if you wish. However, this is a humbling activity -- your odds of success are extremely low. At least you are doing something for yourself, though.
I think my last piece of advice is one my father gave me when I was young. He said that he always wanted to be a scientist when he grew up and so he became a university professor. However, as he got older he realised that there were a whole bunch of jobs that he would have really enjoyed. It didn't really matter which one of them he picked. He enjoyed his career, but there he would have enjoyed many others as well. If programming is not working out, find another career. There are lots of them and they are all rewarding in their own right.