My intrepid adventuring character has been attacked, and the damage reduced him to 0 hit points and falls unconscious. On his next turn, he rolls a Death Saving Throw, and gets everybody's favorite die roll: a natural 20! He regains 1 hit point, and thus regains consciousness.
Now that he's conscious, can he still move and take an action for his turn? The rules say that you roll a Death Saving Throw "Whenever you start your turn with 0 hit points," but it's not clear to me if that's replacing one's normal turn, or if it's just at the start of one's turn, and the only reason one normally doesn't follow it with a movement and action is because unconscious creatures are incapacitated and can't move.
There's some discussion in the comments of an answer to "Do you regain consciousness immediately on a 20?", but it doesn't seem to have an authoritative referenced resolution to the discussion, and it seems that if it's worth talking about in comments it's worth asking as a full-fledged question.
The entry for Death Saving Throw says:
So this just means that you make the throw at the beginning of your turn. However, nowhere in the description does it say that it requires any kind of action to make the save.
There is no general rule saying that doing any kind of saving throw (of which a death saving throw is one) takes any kind of action at all. Indeed the quote above would imply that it is not one. This especially makes sense since many saving throws are made not on your turn and thus would be unlikely to use your action or movement or reaction. Additionally, if saving throws did take any kind of action to resolve, it would be quite a huge power boost for those abilities that cause them.
That being the case, once you regain consciousness, you should have all of your move and action available to you to use as you see fit.
A good point from @MivaScott: