The buses from Tallinn to Riga all go along the Riga Bay (Tallinn-Parnu-Ainazi-Salacgriva-Riga) and not through Sigulda.
You can get from Tallinn to Sigulda by train, by taking Tallinn-Valga train (which leaves Tallinn at 6:53am) and then switching to Valga-Riga train (which leaves Valga at 11:43am and stops at Sigulda at 13:49). I think this would take about the same time as Tallinn-Riga-Sigulda trip by bus, because the train is quite slow (it does not use the most direct route between Tallinn and Riga).
the fastest way is to step out of the Tallinn/Riga bus in Saulkrasti and then take a taxi to Sigulda (I'm not sure that all busses passes through Saulkrasti). There are just 40 km distance to Sigulda or 36 km to Turaida (I guess that your destination is Turaida Castle, but not Sigulda).
If you take the taxi and have time, then stop for an hour or two at Birini castle (it may be closed on weekends for private events).
I guess that the taxi would cost you about 20 Lats, but negotiate the price first.
Check working hours of your place of interest you are going to.