One of the neat features of Final Fantasy XIV: A Realm Reborn is that you can be any class in the game and level them all up to level 50. What is even neater is that certain skills can be used across multiple classes, so you can cast Protect and Cure as a Lancer for instance. What is even more awesome is that you get a bonus for leveling up other classes based on your highest class, which makes leveling "alt classes" even easier.
It is definitely one of the features you shouldn't neglect, as additional classes can make life much easier by giving you additional attacks, debuffs, and variety. It also answers that question of "what should I be?" when your friends start to pester you to join their group.
First, you have to have authorization to change classes which comes through during your story quests around level 13. When you're told that you can change classes, quests pop up in the three major city-states that allow you to switch classes and start crafting (as crafting professions are classes).
To unlock a class you'll need to complete a short introduction quest, at which point you'll be given a weapon appropriate for that class and sent on your way.
To change classes, simply equip the unique weapon for that class.
First, pick your primary class and switch to it. Next open the action and traits window and select the class (below the word additional) that you have unlocked and want to use a specific skill from. A list will appear showing which skills your current class can utilize. You can then drag the skills to the bar at the bottom of that window (additional actions) to equip them to your "shared" bar. This starts a lengthy cooldown and then the ability can be used.
You can use the ability by dragging it onto your hotbar / action bar and then well, use it!
Jobs can only utilize a very small number of other class skills, usually from only the two classes that were required to level for them. For instance, a Bard can only use skills from Pugilist and Lancer. So if your aim is to be a specific job, then understand that you can only use skills from those select classes.
Well, for non-job restricted classes, I would say leveling up a Conjurer is a good idea for Protect and Cure. For melee classes, you're going to want to level up a Lancer to level 2 for Feint to give yourself a snare. For casters, be sure to get Cure to help heal in battle. Protect is just good for everybody.
Be sure to check the tool tips to see what skills can be used where. Balancing patches may alter things a little bit, so being aware of things like that is important. You also should work out a plan on how to level the different classes up, taking an easy to level class like Lancer to the highest level first for instance to make leveling other classes easier. Likewise, if your primary class is a job, focusing on the two component classes should be a high priority first.
For leveling up characters quickly, my best advice is just to do the starting class quests and then kill enemies a bit higher than your level and do any nearby FATEs. The first bit is the hardest since you're going to out level most of the gear in your inventory rapidly, but you have to be creative on getting those first levels. Once you can start taking on Guildfates you're pretty much golden.