By far, the best option with Transmogrify is to flip estates, silvers or other cards into a village or draw so you can kick off mid-game engine turns with lower component densities. A hand of Silver, two coppers, Tr, and Smithy is vastly worse than TrTrCCSmithy.
As far as speed at the opening. Meh, it is not blindingly fast like Remake, but it is a pretty nice way to up your cash density. It is a turn slower than Jack, and you have high odds of $3 or lower hands ... but you have a lot of mid-game flexibility. It is an absolute beast with on-trash effects in the early game as you can both improve buying power and trigger your on trash effect (e.g. E -> S, discard Market square, gain a gold). Fortress works very nicely if you have $5 draw and even Squire can work nicely if only to replace the dead Squire. Being able to wait until the best time in a shuffle to trash is very nice if you are doing something complicated. Transmogrify is also very nice on a lot of potion boards, you lower the odds of bad potion hands and also have the option of making the Pot into an immediately playable $5.
The down side of Transmogrify is that you must trash at the start of the turn. If you are building an engine or just have plenty of draw, that makes it ever less likely that you will hit the last estate or that you will find that Province to make into a Plat. This also makes it far weaker at cannibalizing your deck near game end, you have to be awfully sure of yourself to junk a Smithy into a Duchy if you still need to draw a province buying hand to win the game.
One heuristic that isn't bad is to think if you would take Procession here. You do not get the double play first, but you also do not have to worry as much about alignment and you get to play the big card sooner. Something like Transmogrify/Smithy/Bazaar/Grand market/Kc would be ideal. You turn Estates to silvers early and later you chain your way up
Having multiple Transmogrifies lined up is also nice for retooling your deck. E.g. you go Festival/Library and your opponent then goes Council room/Festival, if can be handy to be able to swap Libs for Huntinggrounds or Altars. This can be exceedingly nice when your opponent has committed to a deck that has a particular weakness (e.g. Possession against Madmen decks, discard attacks against a lot of Golden decks, Swindler against Hparty decks). In general, though, I have not seen too much call for multiple Transmogrifies outside of a handful of power combos (e.g. Transmog/Fort/Catacombs, Transmog/Scout/Rabble).