The First Doctor

When The First Doctor enters, search your library and/or graveyard for a card named TARDIS, reveal it, and put it into your hand. If you search your library this way, shuffle.
Whenever you cast a spell with cascade, put a +1/+1 counter on target artifact or creature.

The Thirteenth Doctor

Paradox — Whenever you cast a spell from anywhere other than your hand, put a +1/+1 counter on target creature.
Team TARDIS — At the beginning of your end step, untap each creature you control with a counter on it.

TARDIS

Flying
Whenever this Vehicle attacks, if you control a Time Lord, the next spell you cast this turn has cascade and you may planeswalk.
Crew 2 (Tap any number of creatures you control with total power 2 or more: This Vehicle becomes an artifact creature until end of turn.)

The Fourteenth Doctor

When you cast this spell, reveal the top fourteen cards of your library. Put all Doctor cards revealed this way into your graveyard and the rest on the bottom of your library in a random order.
You may have The Fourteenth Doctor enter as a copy of a Doctor card in your graveyard that was put there from your library this turn. If you do, it gains haste until end of turn.