Having a class skill means it's cheaper to improve the skill when you level up. They don't normally give you bonuses or anything, just permit gaining a full rank per point spent, instead of only the ½ rank per point spent that you get when buying cross-class skills. Because class skills are a permission, they don't stack — having a class skill from two different sources is just a redundancy, not an advantage.
Class skills matter most when you level up, since that is when you're spending your skill points to improve skills.
Regarding class skills attached to classes: When levelling up, only the class skills of the class that you are currently gaining a level in count. That means that if you are gaining a level in Infiltrator, but your have previous levels in Fast, you ignore the class skills that Fast lists and reference only the class skills of Infiltrator. Class skills for Fast that are not class skills for Infiltrator are purchased as cross-class skills (that is, you only get ½ ranks), despite having previous levels in Fast. From the SRD, emphasis mine:
When a multiclass hero gains a level in a class, he or she spends that level’s skill points as a member of that class. Only that class’s class skills may be purchased as class skills. All other skills, including skills for another class the hero has levels in, are considered cross-class skills when ranks in those skills are purchased at this level.
Class skills granted by Occupations are an exception to all this: they are explicitly permanent, and always count when you are levelling up. They also explicitly give you a +1 Competency Bonus if it gives you a class skill that your 1st-level class also has. The former means that, no matter what the class skill list for the current class being levelled, when you spend skill points on your Occupation-derived class skills they are always improved by a full rank per skill point spent. The latter means that a Transporter occupation (which make Drive a class skill) combined with a Fast Hero class at 1st level gives a permanent +1 competency bonus when rolling Drive checks. This bonus is regardless of later class levels, as only the 1st-level class matters for it.
An example of buying class and cross-class skills when multiclassing
For example, let's say you're a Transporter who has 3 level of Fast Hero and you've just earned enough XP to advance to 4th level. You want to take this level in Infiltrator, to become a Fast 3/Infiltrator 1. Let's say for simplicity your Int bonus is +1, so with the 7 skill points Infiltrator gives you, you have 8 to spend:
- You want to improve Craft (mechanical) by two ranks, which is a class skill for Fast but not Infiltrator. That costs 4 skill points because it is a cross-class skill for Infiltrator and you only get a ½-rank per skill point for cross-class skills.
- You also want to improve Hide by two ranks, which is a class skill for Infiltrator and Fast; because it's a class skill for Infiltrator that costs 2 skill points (and being a class skill for Fast is irrelevant).
- Lastly you want to improve Drive by two ranks. This is not a class skill for Infiltrator. It's a class skill for Fast, but that doesn't help you, because right now the class skills for Infiltrator matter and the ones for Fast don't. Normally, improving it by two ranks would cost 4 skill points (more than you have left) because it is a cross-class skill for Infiltrator, but your Occupation is Transporter, and that makes Drive a class skill for you regardless of class. Because of Transporter, you only need to spend 2 skill points for two ranks in Drive.