Financing your graduate education with a teaching assistantship is a fantastic experience. As a TA I learned that I really liked teaching, which was a surprise, and I am still teaching today. Even if you don’t end up teaching, you will increase your comfort level in public speaking, which is helpful for any career. Plus by teaching a subject you will learn it at a deeper level.
So if I convinced you to try to become a TA, here’s my advice on finding a position:
1)When you apply for graduate school, e-mail professors in your department and let them know you are interested in a TA position in their courses. Often professors pick their own Teaching Assistants for their courses.
2) Let your department chair know that you are interested in being a TA. Sometimes they have the job of filling last-minute positions that have opened up.
3) If you are not a native English speaker looking to work in the US, work on your English and take the TOEFL test. This is required at most schools for foreign students.
Keep in mind that you WILL NOT make much money in these positions. Research positions often pay more. BUT you will be getting valuable experience.
More importantly, all TA positions typically carry with them a full tuition waver. Not only will you get free tuition, but also all the perks of being a full-time student, like free access to the gym and to cultural or athletic events. So even if the pay is poor, the fringe benefits are excellent.