Hello! Does anyone know whether it's OK to use the same links in similar but different web sites? I'm pretty new to SEM and I don't want to get dinged! Thanks!

Welcome Kristin,

similar but different
"Different" is the operative word here. As long as the sites are different you shouldn't have any problems. One good way to differentiate sites is to have a different backlink profile. So here, where the forward link profile might be very similar, different incoming links should help make the sites more different.

Are you asking about repeated anchor text?
If yes I would not do them all the same... try synomyns but some repetition is good. Mix them up.