Your urge to become a research assistant should be driven by the interest in the topic of research. Once you have the interest, everything else will fall into place. I will be dividing this section in 3 parts:

Before Applying
It is always good to be prepared before you actually apply for the job. Here are a few things that you should pay heed to:

Right Resume
Your resume should showcase all your merits, and it should also specify any previous experiences in the field of research, along with your academic details and achievements. Undergraduates will obviously lack experience, if you are one, then highlight your interest to be part of the research, but you should always remember one thing, there is no replacement for experience.

Job Search
Your success as a research assistant is entirely dependent on the topic of the research. So, always assist or participate in research that interests you. This interest will keep you motivated, in times of stress. Always begin your job search from the department notice board. Ask your professors for any vacancies, they can be very helpful. Most of the times, the openings for a research assistant are never advertised, so you should always stay in touch with your faculty members. If you are a graduate, or someone with advanced degrees, then you can apply in various companies that carry out research. You can use the web, and the local media to find openings in such companies. Do not forget to include a research assistant cover letter, when you apply.