If I search for a Miracle and put it in my hand, can I cast it for its Miracle cost?
No. The card changes zones but is never drawn by you.


  120.5. If an effect moves cards from a player’s library to that player’s hand without using the word
  “draw,” the player has not drawn those cards. This makes a difference for abilities that trigger on
  drawing cards and effects that replace card draws, as well as if the player’s library is empty.


This means you can't cast the cards you tutor directly into your hand by paying their miracle cost.

It does, however, mean that any cards you tutor up with Mystical Teachings won't count towards your "first card drawn this turn" -- so you could use Teachings on your upkeep and still cast a miracle card you drew naturally, for example.

If you want to cast miracle cards with tutors, use ones that put cards on top of your library, like Mystical Tutor and Liliana Vess.