There's quite a few places that you can learn Dovahzul, I'd recommend trying each and seeing what works best for you.

However, you should never use the translator to learn Dovahzul. As it takes the literal translation, which is often wrong.

There are lessons on Thuum.org , which cover all the basic aspects of Dovahzul, it includes excercies, examples, and a few tools to help you learn.

There are also Memrise Courses, we have the canon course and the non-canon course. The difference between is that canon words are from Skyrim itself, and non-canon words have no relation to canon words. Memrise teaches through repitition, using online "flashcards" and excercies such as "choose the translation".

Other members, such as myself and the newly promoted moderator Liis, prefer to learn by learning sections of the dictionary and memorising the affixes and grammar.

If you have any questions about Dovahzul, either create a thread in the forum (Dragon Language Discussion) would be best for Dovahzul-related questions) or ask on the community wall.