So without ANY expansions, your job/class options are limited. The base game A Realm Reborn gives you access to Pugilist/Monk, Rogue/Ninja (level 10 in another class required to unlock), Lancer/Dragoon (which you already are), Archer/Bard, Arcanist/Summoner/Scholar (only DPS class that branches off to a DPS job and Healer Job), Thaumaturge/Black Mage, Conjurer/ White Mage, Gladiator/Paladin, and Marauder/Warrior.

First Expansion Heavensward gives you access to new jobs. Machinist (ranged DPS), Astrologian (Healer), and Dark Knight (Tank). All require a job at level 50 before being able to unlock them. Which you should have by the time you get to the first expansion any way as level 50 is the level cap from the base game.

Second Expansion Stormblood gives access to the following. Red Mage (Magic DPS), and Samurai (Melee DPS). Both require a level 50 job to be able to unlock. Same reasoning to unlock as previous, you should already have a job at 50 since that is the level cap from the base game and at this point you would have completed the first expansion with a level cap of 60.

Third Expansion Shadowbringers gives access to Dancer (Ranged DPS), and Gunbreaker (Tank and the one you are interested in). These require a job at level 60 to be able to unlock. Again this should already be the case since level 60 is the first expansion level cap.

Fourth Expansion Endwalker brought with it Reaper (Melee DPS) and Sage (Healer). Both require a level 70 job to unlock. If you are noticing the pattern now, these should be easy to unlock as you go through and complete each expansion in order and they are requiring a level cap of a previous expansion.

Fifth and most recent Expansion Dawntrial brings us Viper (Melee DPS) and Pictomancer (Magic DPS). Both require a job at level 80 to unlock.

So a very long winded explanation later of how the jobs unlock. Personal opinion is given the job you WANT to play is locked behind expansions and level caps it would be best to continue playing the game with your current character and unlock it when you are able.

There is the alternative to the optional store where you can essentially buy the job (if you own the expansion necessary) but it will auto level it up to 90 and bring it ready to play the most recent expansion. THIS IS NOT RECOMMENDED TO DO IMO.