Can Gladiators synthesize stuff?

To synthesize you need to switch from your Gladiator profession to the Alchemist profession by taking use of the armory system, which is achievable by switching your primary weapon with an alembic.

After you reach level 10, the Gladiator guildmaster will have your level 10 class quest for you. Upon completion you will be able to unlock other classes.

To unlock a class (regardless of whether it is battle, gathering or crafting) you go to the respective guild for that class (in this case, the crafting guild of your choice) and take the quest to become a member.

Early on you only have easy access to the guilds within your starting city, but around level 15-17 depending on your main story quest progression you obtain an airship pass which gives you easy access to the other 3 cities.