Stay somewhat detached. At the best of times, children and parents have complicated relationships. Don't get too involved in their problems or issues. Respect your partner's relationship with the children, whether you agree with it or not. Discussing the children with your partner can be tricky, and if a child trusts you enough to confide in you, the reverse will also hold true.
Nobody likes a phony.
As with getting to know anyone, showing a genuine interest in the children will make your relationship more satisfying for all of you. The children may indeed become new friends. It's important to keep your boundaries clear. Although you may grow close, you will never be the child's peer. Your primary relationship is with the parent.