Research Tamil classes in your area. If your area has a significant Tamil population, there may be classes available at a local community center or through a private tutor. Some universities, such as the University of California Berkeley, offer Tamil courses.
Use Internet resources. There are many free places online to learn Tamil. The University of North Carolina has an extensive lesson database for would-be Tamil speakers.
Spend a lot of time on the Tamil alphabet. Learning the writing system is one of the largest hurdles you will have to overcome to learn Tamil. Practice writing, but also practice reading out loud from Tamil texts to improve your comprehension.
Find a language partner. If someone else in your area is learning to speak Tamil, try to arrange meet-ups where the two of you can practice. If there's no one nearby, the site Live Mocha can connect you with other students of Tamil for conversation practice.
Immerse yourself in the Tamil language as much as possible. If you live in a place where no Tamil is spoken, look for online radio in Tamil. The BBC has a Tamil homepage with many visual and audio resources.