Timey-Wimey Car Crash
There is no initiative. When I deliver a post declaring the 'Player Phase' of a round of combat, that signals the start of a 24-hour period for the players to post their characters' actions for that round.
Events in a round all happen at once, like a car crash. Rather than one car hitting another, and then the second car responding by hitting the first, they hit each other at the same time. In game terms, this means that everyone deals damage (and heals) at the same time.
For example, if a fighter and a goblin attack each other in the same round, there are a couple of possible outcomes. Lets say the fighter rolls to attack, hits, and deals enough damage to kill the goblin. What happens if the goblin also deals enough damage to kill the fighter? Like in any number of real-world examples, both combatants might die.
This means that any combatant present and active in a fight at the beginning of a round is guaranteed their turn that round.
Don't Cross The Beams!
There is a necessary degree of separation between what the combatants can do to affect each others' actions in a round. Since the turns in one round can be posted in any order, it all gets really confusing if the player controlling the fighter makes their attacks and rolls their damage, and then later that day the player controlling the cleric casts Bless. Does the fighter go back and retroactively add the bonus to the attack rolls? Nope.
Any spells or effects you do that affect other people in the game (besides healing) don't have any mechanical presence in the combat until the end of the round.
For example, if the cleric casts Bless on the fighter during Round 1, the fighter doesn't benefit from it until Round 2 starts. If the cleric instead casts Hold Person on the goblin during Round 1, the goblin's saving throw happens at the end of the round, and if the spell sticks then the goblin starts Round 2 paralyzed.
This also means that any status effect in place on a combatant at the beginning of a round is in effect for the whole round.
Same-Round Healing
Healing happens at the same time as damage.
For example, if in Round 4 the fighter and goblin both deal enough damage to kill the other, and the cleric casts Healing Word on the fighter, the fighter receives the healing at the same time as the damage, possibly keeping him on his feet instead of succumbing to the blow.
You Are Special
Effects you create that affect yourself happen immediately, and you benefit from them just as you would in a regular combat.
For example, you can cast Hex on an enemy and then attack them that same turn and deal your extra damage.
However, some spells and effects are more complicated. For example, you can cast Invisibility on yourself to immediately be able to get away from enemies without provoking opportunity attacks from them. But for the turns of those enemies that round, you are still considered visible until the end of the round.
Real Example
An example of all this in action can be seen in this combat from my Icewind Dale game.