So, let's say I'm multiclassing Barbarian and Fighter and I take climb as a skill. Would I get a +3 from both classes, or only from one?
No, the Class Skill bonuses are a one-time-only (per skill) thing.1 See the PFSRD:
In addition, each class has a number of favored skills, called class skills. It is easier for your character to become more proficient in these skills, as they represent part of his professional training and constant practice. You gain a +3 bonus on all class skills that you put ranks into. If you have more than one class and both grant you a class skill bonus, these bonuses do not stack.
Multiclassing does give you the +3 bonus to any skills you already had, but had not previously been a class skill, however.
Really minor footnote, probably not worth reading:
If you select a level in a new class, all of its class skills are automatically added to your list of class skills, and you gain a +3 bonus on these skills if you have ranks in them.
1 Super-strictly speaking, you do have the bonus twice, but since they are equal and do not stack, that will never become relevant (i.e. you have a +3 bonus and a +3 bonus, and have to pick just one – it doesn’t end up mattering which you pick).