D&D 5e's initiative system is more-or-less unchanged from much earlier editions. Every character has an "Initiative Bonus"; at the start of combat everyone (including all the DM-controlled enemies) rolls a d20 and adds their initiative bonus; then everyone takes their actions in descending order of their rolls. When everyone's gone once, it "goes back to the top" and repeats in the same order.
This... works. It gives you an ordering and lets you represent a faster character by giving them a higher initiative bonus... sorta. But it has several problems.
First, your initiative bonus just doesn't matter that much. A d20 has a lot of variation. Over the course of many rolls, you can distinguish between, say, a +2 and a +5 bonus in how many times you succeed. Initiative simply isn't rolled that often, tho, so a character with +5 to initiative won't feel like they're actually much faster than a character with +2 to initiative.
Second, in practice it's a rather slow, clunky way to start a battle. A perhaps dramatic build-up to combat suddenly screeches to a halt as the DM demand initiative rolls from everybody, rolls a bunch of initiatives for their monsters, and then sorts everything out. This can easily take several minutes! (It doesn't seem like it should - it sounds easy and quick - but theory and practice don't align well here. In practice, it's pretty slow.) Only after all that's done can combat, and fun, actually begin.
Third, once the initial initiative roll has happened, and the first round has finished, initiative... doesn't matter anymore. The order just determines who gets to strike first; after that, every round is the same for everyone: you go, then everyone else gets a turn, then you go, etc.
Fourth, while players aren't technically locked into their initiative result (they can delay and take their turn later if they need to), in practice players don't (for various practical reasons). This restricts what sort of combos people can use; it might be more effective to let the Fighter rush forward and have the Cleric hold back to see if they need to drop some heals or just do cleanup, but if initiative puts the Cleric first, generally they'll just go first. This can get very frustrating!
This article presents a better version of initiative, that both simplifies things and gives players more meaningful options. Their write-up didn't handle some corner cases well, tho, so I've reproduced and cleaned up the idea for my own purposes:
The core idea is that initiative is done away with. Instead, each round, players announce whether they'll be taking a "fast" or "slow" round. All the players taking a fast round take their actions immediately, in whatever order they decide amongst themselves.
Second, the DM decides which monsters are taking the "fast" or "slow" round - fast monsters take their turn now, in whatever order the DM wants. (Typically, all the "mooks" will go in the fast round.)
Third, all the players who chose to take a "slow" round take their turns, in whatever order they wish. However, because they held back, examining the battlefield and waiting for an opportune moment, they can add advantage or disadvantage to a single roll anyone makes during their turn. (They can give themselves advantage on an attack roll, or give an enemy disadvantage on a saving throw, or provoke an Opportunity Attack and give the enemy disadvantage on their attack, etc.)
Fourth, the "slow" monsters take their turn, and also get to impose advantage or disadvantage to one roll during their turn. (Typically, the "significant" enemies will go here.)
Then the round is over, and the next round begins, with players once again choosing to go fast or slow.
That's it! (Except for some of the additional quirks, noted later in this post.)
In practice this ends up having a lot of benefits over traditional initiative.
- Because there's no big "initiative list" setup at the beginning of the combat, you can jump straight into combat with no delay. Just ask the players who's going fast, and you're off to the races. This has a surprising psychological effect on players, maintaining the drama that was built up pre-combat very effectively!
- Because the players can adjust when they take their action each round, they remain engaged thru more of the round, rather than just perking up on their turn and checking out a bit while they wait for everyone else to go. They plan out their actions along with the rest of the party, setting up combos and adjusting things for optimal safe ordering. You end up getting a lot more interesting teamwork out of people as a result!
- It's so fast! Even on a round-by-round basis, this really does make combat move faster. Because the players are working together and going all at once, their plans don't collapse as much due to enemies taking actions between them (and players don't simply forget what they were going to do, which is a significant danger normally...). As such, players don't have to reassess the battlefield before each of their turns - they know exactly what's changed, since it just happened and was part of the plan.
- No more (or at least, much less) forgetting about people! It's remarkably easy to occasionally skip people in the initiative when using it normally; if a non-active player asks a question, it's easy to slip back into the order as if they'd just gone. Since the players and enemies all go in just two large groups, tho, it's much simpler to track everyone - the players will remember themselves, and enemies become dramatically less fiddly to track.
- Slow rounds are amazing for players who want to get off a big dramatic action with less chance of whiffing. Similarly, they're great for making your Big Bad actually threatening, rather than several rounds of "They swing, and... they miss. Again. Your turn."
While the core rules above are trivial, there are a few additional details to cover.
First, several classes or feats give bonuses to initiative, which no longer do anything. (Alert feat gives +5, Revised Ranger gives advantage, etc.) While initiative bonuses aren't actually very significant, and thus it would probably be okay to just drop them, players don't like losing abilities even if they're minimal, and it's still a cool differentiator for a "fast" character.
As such, any ability that grants a "significant" initiative bonus (+2 or higher, more or less, but use your best judgement) is reinterpreted to let you get the slow-round bonus (adv or dis on one roll during your turn) during a fast round once per long rest. If you have multiple sources of bonuses, they stack to give you multiple uses of this ability.
The Bard's Jack of All Trades and the Champion's Remarkable Athlete don't count; their bonuses only range from +1 to +3 and aren't really "significant", plus most people don't realize they apply to Initiative in the first place (it's a Dex check!), so whatever.
There are some details to work out for spells that last X rounds (particularly those that are "one round") that I'm not sure about.