Proof for Transitivity of Weak Bisimulation ≈
You are on the right track. It can be useful to prove the following lemma: if $\sim$ is a weak bisimulation, if $P\sim Q$ and if you have a sequence of transitions $P\xrightarrow{w}P'$ for $w\in (Act\cup\{\tau\})^*$ then there must be a sequence $Q\xrightarrow{w'}Q'$ with $P'\sim Q'$ and $w =_{\tau} w'$, i.e., $w$ and $w'$ are "the same modulo $\tau$". This can be done by induction on $w$, and it will come in handy when you deal with transitivity. Check that $u=_\tau u'$ and $v=_\tau v'$ imply $uv=_\tau u'v'$, i.e., that $=_\tau$ is a congruence.