I think you should be at least level 68, and you require an alchemy skill of 325+.
To become a Master you need to do a quest.
For Master of Potions, go to Lauranna Thar'Well in Zangarmarsh (Cenarion Refuge).
For Master of Elixirs, go to Lorokeem in Shattrath (Lower City).
For Master of Transmutation, go to Zarevhi in Netherstorm (Stormspire), but I think you need to speak to the alchemist in Thrallmar/Honor Hold first.
For all quest, you'll need to collect some items. The items for Transmution (4x Primal Might) are the most expensive, but don't require you to run instances as is needed for Potions/Elixirs.
Which mastery is best, depends on your personal needs, your guild's need, the prices on your server and your preference. For example, if prices of herbs (= mats) are much higher than prices of elixirs (= product), you might reconsider taking Mastery of Elixirs.