The Thaumaturge is the only traditional damage dealing mage class in Final Fantasy 14: A Realm Reborn. An Arcanist or Summoner can be considered a damage dealing mage class, but they inflict punishment via damage over time spells (DoTs) and by use of their pets. Thaumaturges inflict significant damage in large bursts, especially once they hit level 30 and transition into a Black Mage.
When you first begin the game as a Thaumaturge, you start in Ul'dah where the Thaumaturge Guild is located. At first, you'll only have Blizzard available to you. At level two, you gain the ability to cast Fire, then at level four you learn Transpose, and at level six you get Thunder. At this point, you have all of the main elements to learn how to play as a Thaumaturge throughout a majority of the game.
A Thaumaturge almost never has to worry about MP. This is because casting Blizzard on an enemy creates the effect of Umbral Ice. While Umbral Ice is active, a Thaumaturge regains MP at a high rate. It's basically impossible to run out of MP if you use nothing but Blizzard-based spells. However, Blizzard spells do not inflict a great deal of damage. You can kill an enemy with nothing but Blizzard, but it's not going to be a quick or efficient death.
In order to inflict large amounts of damage with a Thaumaturge, you must use Fire-based spells. When Fire hits an enemy, it removes Umbral Ice and grants the effect of Astral Fire. This increases the damage of Fire-based spells, while also increasing the MP consumed by them. At the same time, it lowers the cost of Blizzard-based spells.
This is where Transpose comes into play. It allows you to swap between Umbral Ice and Astral Fire. Once you cast enough Fire-based spells to deplete your MP, Transpose to switch to Umbral Ice and regain your MP and use Blizzard-based spells, then Transpose again to go back to Astral Fire and continue inflicting damage with Fire-based spells.
In addition to Blizzard and Fire, you should use Thunder. This is a Thaumaturge's primary DoT spell, and should be cast at the beginning of every fight. At level 28, you learn the Thundercloud trait, which grants a five percent chance that with every damage over time tick of Thunder, the next Thunder-based spell will inflict all of the DoT damage at once (in addition to the continued tick damage), and cost no MP.
While five percent may seem like a low figure, when you toss Thunder on multiple enemies and factor in a tick roughly every three seconds, that's a minimum of four ticks for every Thunder I spell used (more for Thunder II and Thunder III). On just three enemies, it's not uncommon to see Thundercloud proc quite often.
Your basic rotation as a Thaumaturge should be to start each fight with Thunder, then use Fire spells until you're low on MP. At this point, use Transpose to switch to Umbral Ice, then cast Blizzard spells until you have full MP again. Once that happens, Transpose again to switch to Astral Fire, then repeat the process. If Thunder runs out and the enemy still has a good amount of health remaining, use another Thunder spell right after Transposing to Umbral Ice (so you're regaining MP while casting Thunder).
When you make the transition to Black Mage, a few things change. At level 34 you gain Fire III, and at level 38 you gain Blizzard III. While Fire II and Blizzard II are area of effect spells (AoE), the third version of these spells grants full stacks of Umbral Ice or Astral Fire. The more stacks you have, the more damage you inflict or the faster your MP replenishes. Once you have both spells, you no longer use Transpose unless you end a fight in Astral Fire, in which case you Transpose to switch to Umbral Ice and replenish your MP.
With both Fire III and Blizzard III, your rotation changes to: Thunder (the length of the fight determines which Thunder spell to use, as each spell lasts a different amount of time) > Fire III > Fire (until your MP is low) > Blizzard III > Thunder (if it has worn off) > Repeat. The only change to this comes at level 44 when you learn the Firestarter trait. This grants a 30 percent chance that a successful Fire cast will make your next Fire III require no MP and cast instantly. When this happens, instead of casting another Fire spell or Blizzard III, you cast Fire III.
When you reach level 50, Black Mage learns the spell Flare. This spell requires a minimum of 266 MP and consumes all of your MP, but inflicts more damage than any other spell. It's also an AoE spell, meaning it hits multiple enemies in close proximity. It's typically used at the end of a rotation, before switching from Fire to Blizzard III. However, because of the casting time of Flare, it's generally more effective to use it only when hitting multiple enemies, or when you have the Convert and Swiftcast abilities available. If you leveled an Archer, Raging Strikes is also very effective when used in conjunction with Flare.
A typical Flare rotation would look something like this: Thunder > Fire III > Fire (until MP is close to 266) > Raging Strikes > Swiftcast > Flare > Convert > Flare > Transpose > Blizzard III. Raging Strikes increases damage dealt, making Flare hit harder. Swiftcast allows Flare to cast instantly, negating the lengthy cast time. After the first Flare cast, use Convert to restore 30 percent of your MP so you can cast Flare again while Raging Strikes is still active. After the second Flare, use Transpose to switch to Umbral Ice and begin replenishing your MP, then cast Blizzard III to continue inflicting damage while giving yourself more stacks of Umbral Ice to replenish your MP faster.