Do *NOT* go to the TTC. There is NO bus to Downtown Disney or Typhoon Lagoon there.
The bus to Downtown Disney/Typhoon Lagoon will pick up right outside the Great Ceremonial House. (Be sure to read the signs as to where the stop is... they are currently building a new bus stop)
As for how long it takes, that can depend. Often times, morning routes will go to TL first, then DTD. Afternoons, DTD then TL. Shouldn't be more than 20 mins.