As others have said, it will help a lot if you take it easy with the low level dungeons first as a new player. The higher dungeons will start to expect healing cooldowns that use no mana, but restore a decent amount of health. A certain gear item level is also expected, so we need to make sure we get new gear whenever we can. Try to aim to keep the gear around your class level the best you can.

If it's possible, maybe try out the Conjurer / White Mage class for a bit to get comfy with the basic spells first. Satasha will start off with mostly using just Cure and Medica (or Benefic and Helios with Astrologian). I believe the cooldown Lucid Dreaming comes up fairly soon and that one is important to use when you drop to around 80% of your mana or lower. It contributes a lot of mana regeneration, so make a habit of using it whenever you can. Then when you get to level 30, don't forget to visit the class guild again to upgrade your class to a job class. It then sends you on job quests to earn spells that will be important to use. An example is Regen that appears for White Mage at level 35. That one gets put on the tank(s) as often as possible after they finish pulling enemies to heal them over time (a HoT for a short term). Try your best to wait for the tank to finish pulling before using the HoT heals or the enemies might potentially start chasing you if the tank accidentally didn't hit them all yet. Astrologian can use the Diurnal Sect buff and Aspected Benefic to get a similar regen too.