The job market in LA isn't good, for any profession. Maybe call this organization and ask:
LA Paralegal Association http://www.lapa.org/
LA law work is tight right now, so many attorneys are keeping hold on any work that comes in, and cutting their rates, so they can make their billing hours. Paralegals were a good profit maker for the firms, as they would bill out paralegals' time at much higher rates than they paid them. But right now the attorneys need to keep on to those billing hours themselves.
Paralegals can spend a lot of time doing document review, which is very very tedious. At every firm my dh has worked for, that is basically all the paralegals do, document review, except for a couple very experienced paralegals who were a bit more help with research, etc. Basically, paralegals do the most boring stuff that the attorneys don't want to do. Maybe at a small firm the work would be more interesting, since they'd likely do more than just document review, but the pay would likely be lower.
You might also look into being a legal secretary. At my dh's firm they make as much or more than the paralegals and they leave the office at 5pm. (The secretaries with many years experience at biglaw firms make $80,000+.) It's also a tight market, with many attorneys now doing much of their work due to computers, and the secretaries work for more attorneys than in the past, but their work load isn't really much more than in the past. While being a legal secretary takes specific skills and knowledge, a legal secretary is also employable in other types of offices, whereas being a paralegal isn't so flexible.
Usually, firms will want to hire those with a few years of experience, so it might take a lot of searching to find your first job.
See if you can take classes at a community college, where the costs are reasonable.
As an example:
PCC Paralegal Studies http://webcms.pasadena.edu/display_program.asp?program_id=330
Good luck!