Update: Since a lot of people seem to come across this page looking for a leveling guide (sorry, mine isn’t finished yet), here’s a link to one that does a decent job at showing you how to reach the level cap quickly with any class or job.
To join other guilds, simply go to their guild house and talk to the guild register who is usually behind the counter. He will ask if you want to become a member. Say yes, and you’ll be given a weapon to activate that class. To now switch between the classes, you only need to change your weapon and you can do this quickly by saving your gear sets and hot keying them, allowing for instant gear change. However, swapping classes puts a cool down on all of your skills. You’ll also want to note that even if you hit level 10, you’ll not be able to join the other class guilds until you gain access to their major city by completing the level 15 story quest. For example, a level 10 Gladiator that starts in Ul’dah cannot start leveling his Conjurer until he gets access to the Gridania city at level 15.
There are two things to note about swapping classes. One, the attribute points earned at level 10 and then every subsequent two levels are locked to that ffxiv class; meaning you can spend all your points in mind as your Conjurer and then put all your points in dexterity as your Archer and they will not be shared. Two, you can borrow skills from other classes as long as they have the correct affinity to that class. For example, the Conjurer spell Cure has affinity to all other classes, thus when you are playing as another class, you are able to borrow that Conjurer’s spell. Picking the perfect classes to level so that you can get their affinity spells and a nice combination takes a bit of testing things out and allows for a lot of customization.
As for the class acronyms on the affinity list, this is what each class acronym is here. Now you’ll notice on this list some of these acronyms are not any of the eight classes I mentioned, but they are in fact jobs.