You mentioned not being sure about how to switch - and this has ended up really long lol - but hopefully at least some of it will be useful!

Once you hit level 10, you get access to the Armoury System. And basically all you need to do is equip a different class' weapon, and you will switch class. When you switch class, you start back from level 1 as if you had never been another class. So if you have allocated any stats (Strength/Vitality), they do not count on your new class - so you can allocate them more effectively on that class (I.e. for Gladiator go VIT and for LNC go STR).
But there are certain abilities that you can use without being on that class - called cross class abilities. You can tell what these are in the ability screen - they're available for a lot more classes than just GLD & PLD (which is what most of yours will say). And they are usuable at any level. So for example, if you are a level 8 GLD, you get Flash. If you then change weapon and go to a level 1 MRD - you would be able to set Flash as one of your cross class skills - and you do that by going into your Abilities screen.

As a class (Gladiator, Marauder etc), you get access to an extra cross class slot every 5 levels - to a maximum of 10, and can draw from anything. Whereas jobs (Paladin, Warrior etc), only get a maximum of 5, and can only draw from certain ones - Paladin is Marauder and Conjurer?.

The joy of FFXIV is that you can play any and all classes in the game without any penalties (expect perhaps Summoner/Scholar). In fact you get a bonus if you have many classes. And you get to experience from lots of different points of view (Tank/Healer/Damage Dealer). But there aren't many skills that you absolutely require to do your job, but a lot help, and more are expected at level 50.

An absolute minimum of levelling another class will be your requirement for your Job - to unlock Paladin you need to have Gladiator to 30 and Conjurer (I think?) to 15. But you probably don't need to worry about that until getting close to 30.