I see in some forums how complex initiative can get.
I prefer a more simple approach myself. To me, initiative is a process which includes surprise and declarations/intentions of attack.
1) First, roll for surprise if it is applicable. After checking each player/monsters surprise adjustments, then surprise is rolled for each "team".
2) After surprise is determined, I have each PC/monster describe their intentions for this round in terms of what weapon/spell they are preparing to use. what change of position they will be making and if there is a specific target they are aiming for.
1d6 is used by each "team" PC's vs monsters.
3) It's then a "roll off" to see who attacks first (provided they aren't surprised). Highest number takes initiative and in the case of a tie, roll again till a winner (high number) emerges.
Now we know if and who is surprised or not. We also know who has the upper hand going into battle.
The next obvious question is "who is attacking who?"
I allowed each PC to identify their target if they wanted to after the surprise roll. What the players don't realize is as they were declaring their intentions, the monster(s) in my head, if they are intelligent enough to, were also declaring their intentions and targeting a specific PC as well.
If the monster is not an "intelligent" monster or the PC doesn't really know who or which monster they are attacking, essentially just running into battle eager to get to the first opponent within reach, I assign a number to the potential targets. if there are say, 10 monsters and the PC did not declare a target, I grab 1d10 and mentally assign each of the monsters a number one through ten. Roll the dice and there is the intended victim.
Same goes for PC's when it's the monsters turn to attack. Unless a specific PC is identified, say a team of four PC's is in action, then I grab 1d4 and mentally assign each PC a number one through four. Give it a roll and there again is the intended potential target.
Sometimes you have a number of PC's or monsters that don't exactly match none of the die you have available. Say you have 3 PC's and the monster is just going in randomly. I will take the die closest above the number the number of targets and re-roll on the un-used number. OR, if it is a team whose numbers are divisible, say two, then I grab a die, like d4 and every other number relates to a target. say one and three relate to PC 1 and two and four relate to PC 2.
Another hotly discussed area of initiative and melee is spell casting. The rules can seem a bit complex in the PHB on this. Personally, I take the easy way out and if the initiative dice of the attacker was the winner, then they will interrupt the spell casting declared by the spell caster. Just like that.
No determining this on that segment or turn or whatever. If the opponent lands an attack and they had rolled initiative, consider it done.
I look at it this way. This is supposed to be a game, not a historical battle re-enactment. Not only do we want to play the game, we also want to keep it moving.
The above is about the extent I go through in melee fighting. It's quick, it's random, it's not all that complex. I generally let the dice determine those things until a player gets out of order. They become cocky and reckless or they are oblivious to the game because they are being distracted by a tv or something. Suddenly, the dice don't mean as much anymore and I start getting creative, making the calls myself.
Enjoy the game.