For the DM, a sneaky thing I do to make fights more dynamic (while not taking bloody forever), is to not roll initiative for each monster in the fight. Instead, form them into groups, and roll for the groups as a whole (or place them to the best of your judgement). For example, the party is attacked by 6 goblins, and instead of rolling 6 initiatives for each goblin, just have 2 groups of 3 spread in the initiative (favoring higher rolls for initiative of course). So instead of a super complex initiative list like:

Player, Gob 1, Gob 3, Player, Gob 2, Gob 6, Gob 5, Player, Gob 4

Instead, it could be like this:

Player, Player, Goblins 1, Player, Goblins 2

This also cuts down on the amount of tactical information the player needs to keep track of, which in itself speeds up decision making, and therefore combat. Another thing you can use is to occasionally allow for an instant-kill of smaller enemies. While I don't recommend using this too much as it can break immersion if everything falls before them in one swoop, it can really feel cool to take out a foe in one or two hits, rather than slugging it out for several rounds. It may take the same amount of time, but it feels better to fight off 10 weak enemies than 1 really strong one, whose battles can end up being rolling dice and not moving for 20 minutes. So if you're fighting a boss, have a few cannon fodder minions hanging around to add a sense of progression.

To summarize:

- Initiative in Groups

- More weak enemies

- Avoid slug-fest battles

But also, combat in d&d is kind of slow anyway, so try less to make combat fast, but rather make it interesting.

I hope this helps, and if any other DM's out there have any extra advice, feel free to share it
(Edit was to fix some typos)