A creature at 0 hp is unconscious and unable to take actions
The rules on dropping to 0hp (found in the Player's Handbook or Basic Rules) say:
When you drop to 0 hit points, you either die outright or fall unconscious (D&DBeyond)
Unconscious is a condition in the game which is defined as:
An unconscious creature is incapacitated, can't move or speak, and is unaware of its surroundings
Being unconscious also means you are Incapacitated, which is another condition. It is defined as:
An incapacitated creature can't take actions or reactions.
Thus, a creature at 0 hp cannot take actions or reactions because they are incapacitated.
There is no current way to heal yourself without actions or reactions while unconscious (without preparation beforehand)
The vast majority of forms of healing that I am aware of (spells, potions, abilities) involve someone taking an action or reaction or bonus action. Since you can take none of those while unconscious, there are very few ways if any for an unconscious person to heal themselves.
Using the spell Contingency 1
Contingency is a spell that allows you to precast a spell and apply a condition that dictates when the spell will take effect. It is a costly spell and takes preparation. But if you used a healing spell with a condition of "when I drop unconscious" or something similar it could heal you while you were unconscious.
There are some specific exceptions to this
A Zealot Barbarian gets the feature Rage beyond Death at Level 14 which reads:
Beginning at 14th level, the divine power that fuels your rage allows
you to shrug off fatal blows. While you’re raging, having 0 hit points
doesn’t knock you unconscious. (Xanathar's Guide to Everything p. 11)
The character with this ability would not be unconscious at 0 hp and could thus heal in whatever way they were able. But this is a specific class ability and is a specific exception to the general rules above.
The rules of 5e have changed significantly from previous editions
As a general rule, just assume that nothing is the same between editions of D&D, it saves a lot of confusion down the road.
The only way to learn 5e is by using the 5e materials. Getting a copy of the Player's Handbook should provide a great resource for learning. You can peruse the basic rules online as well here for free.
1 Thanks @Tenryu.