Finding a regular expression for all non-empty binary strings that contain both 0s and 1s but no consecutive 1s
I would suggest you to draw a small finite state automaton. Things to keep track in the states are "seen 0", "seen 1" and "last letter 1" (to avoid consecutive 1's).

There is an "official" algorithm to go from automaton to expression, but with small examples you usually can read the expression directly.