One of the things I personally struggle with is that pause where the narrative stops and everybody rolls initiative, and then the combat starts.
One tip I’ve used is to roll initiative in advance (at the end of the previous combat is one approach) to avoid that artificial pause.
No offense, but I think that's a terrible idea, as you then totally lose the excitement and tension of everyone wondering what will happen, what they should do, and so on; instead, you get a "pre-scripted" feeling, where Johnny who rolled 20 already knowing they are first, and have their action already optimized and ready to go; Suzzy who rolled lowest tuned out as soon as she rolled initiative half an hour ago so is probably playing on her phone or something waiting for someone to elbow here at the end of the first combat round.
Think of rolling for initiative as the start of a new Chapter in your book - it's a chance to change the tempo, start a new phase, so to speak. The key is to make the change of pace work for your game - the pace of a game should ebb and flow, no-one wants a game that's 100% full-on, non-stop, every minute - you need moments of high action, moments of thinking, and moments for reflection. Like one of my gaming biddies said earlier this year - when he was in the army, they trained to always took half an hour after any fight, because you need some down-time to make sure everything is back in order, to calm down, and generally get yourself ready for the next high-intensity piece of action.
Rolling initiative shouldn't bring your game to a screaming halt, but taking a couple of minutes to say "Roll for Initiative!", then go around the table clockwise / counter-clockwise, asking for everyone's results in turn, and writing them on your piece of paper... it's two minutes well spent. A key part of the "game" in "roll playing game", and it doesn't need to kill the narrative any more than any other D&D rules do.
The keys to success are:
1) making the process of rolling initiative Dramatic, i.e. a deliberate change of narrative pace that says, "now we are in combat mode, game on!"
2) making the process of rolling initiative Quick for everyone - as described above, just ask every player in turn for their number, and write the results down on a big piece of paper, in order from highest to lowest (leave gaps, obviously, to be filled in as you go)