While some moves may be given away during events. Like the grand opening even which gives all characters crated before christmas a free TM/HM/Tutor/Egg/Custom move, you can also train to learn more. How? You can train them with a word count. This largely depends on the rank of the move. The higher the rank of the move, the more training it costs. In addition, you can buy TMs to teach a pokemon. Unlike the games, these are not permanent, and HM's are considered TM's for this site. The cost varies with the rank of the move, as does the word count.
Here are the requirements to learn a move by Word Count:
Rank 1: 200 Words
Rank 2: 400 Words
Rank 3: 600 Words
Rank 4: 800 Words
Rank 5: 1,000 Words
Rank 6: 1,300 Words
Rank 7: 1,600 Words
Rank 8: 2,000 Words