To enter a class for the first time, you must first spend a month of experience as a sort of "entry cost". For each month thereafter, you get certain abilities listed in the chart for that respective level (starting at level 1). Some levels let you chose your abilities, some are fixed. Any time you enter a new class, you must pay an additional entry cost, but if you take the same class a second (or third, or whatever) time (see below), there is no entry cost required.
Most classes have a progress chart for their abilities. Sometimes you get all the abilities for that level, other times you must pick from a list. When you have to chose a number of them, you may chose any ability, spell, or skill from that level OR ANY LEVEL BEFORE. For some abilities it makes no sense to take it more than once, but for the most part, you can take the same ability as many times as you want. Some ability entries have multiple abilities in them (such as "Pas & Speak with Dead"). This means if you select that ability slot, you get BOTH abilities. These abilities represent skills, and choosing this ability is the same as acquiring that skill.
All prerequisites for class abilities are listed in the class advancement chart - you need not worry about the prerequisites for the skill you are taking, just those listed in that ability's slot on the class advancement chart (usually these are the same as the skill prerequisites anyway).
It is possible to be multiple classes, but you can only be one type of magical class (Monk, Mage, or Priest). You can be many different classes of this type (for instance you can be a Battle Priest and a Temple Priest. If you are multiple classes you must conform to all of the limitations. For instance if you want to be a Thief (Max 4 levels in weapons, max 2 points of armor) and a Warlock (max 2 levels in weapons, max 4 points armor), you'd have a max of 2 levels in weapons and a max of 2 points of armor. You can also be the same class twice. But you must fully complete the class before starting it again. Note: if you are multiple Monk classes, your Power Points for each class are separate.