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:
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:
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).