Soundness: $a \vdash b \implies a \vDash b$, i.e. if we can prove something, it will also be true. We don't want a system where we start out with something true and dedice something false. However it is conceivable that even if our system is sound, maybe it's quite incomplete/limited regarding what we can express, which is why we also would like...
Completeness: $a \vDash b \implies a \vdash b$, i.e. if we can show something is true, it's also provable. We want to be able to prove all true statements. However it is conceivable that even though we can prove all true statements, maybe it also proves false ones as well, which is why we'd also like the soundness property from before.
Do I have the right idea?
If so, how would I begin to prove soundness? If we are already given $a \vdash b$ I'm not sure what all we must iterate over in e.g. propositional logic to show that we always get true statements. Especially since it seems possible I could pick a false $b$ that contradicts.