I am currently doing mathematics research as an undergraduate, and all it took was asking my topology professor if anyone in the department would take me under his/her wing over the summer. One of the easiest ways to get into research is to talk to a professor you like and ask what's out there for you.
That said, since you say you've just transferred in the problem is most likely that you don't know any professors at UCLA well enough. My suggestion: give it a year. Most undergraduate research occurs over the summer, and in one year you'll have plenty of time to get to know a few professors. To this end, you probably want to take more than one mathematics course per semester so you can meet more professors.
Use the school year to lay the foundations for a summer of research. Over the school year, the amount of "research" (as in working on some novel result) an undergraduate can do is likely to be limited, and truth be told focusing on your schoolwork is probably best. The general consensus among professors I've gone to for advice is to be well-grounded in the basics - learn to crawl before you try to walk.
However, if you can make time to do some work outside of class, I recommend you look for some independent reading opportunity. Find or have someone suggest a topic that normally wouldn't be covered in class, and either read independently or (my preference) arrange some guided reading with someone. My institution has a program that pairs undergraduate mathematics students with graduate students to read in some interesting topic - you can find the details here http://www.math.rutgers.edu/undergrad/Activities/drp/. This can be a great way to explore your field of interest and find directions for your coursework and future research. (If the topic and pairing work out, this is also a highly entertaining way to do some pretty strange mathematics.)
All throughout, keep in mind the eventual goal: getting a summer research opportunity. If you're thinking about a REU, look for established REU programs early, and think about which ones you'd like to apply for.
I also want to throw in my two cents about your current coursework here (as opposed to making a new comment). If you know the material in the intro to algebra course (and you say you've gone far beyond it), there is no need to waste a semester doing review, unless there is a degree/graduation reason behind it (e.g. required for the major). I find that with mathematics courses, prerequisites are a flexible thing, and what really matters is getting permission from the professor and department. Brian mentioned that UCLA has some stingy policies regarding getting credit by examination, but if the only reason you're taking intro to algebra is to fill a prerequisite then it's probably better to simply request an override and get your credits via some other course.