This might be in the 5e DMG and I'm just forgetting, but I'm a big fan of the 10 minute exploration turn while the party goes through dungeons. I find that it helps things move faster and helps players feel like they're getting enough time in the spotlight during the exploration phase. Rather than figuring out how far they can move in 10 minutes, I just allow characters either to move into an adjacent room (provided there is an unblocked passage to do so) or an action inside of the room. Actions in the room take the whole 10 minutes, but I usually let it slide if a player wants to perform a short sequence of actions to achieve a single result, the whole sequence getting represented by one roll if necessary.
To streamline combat, I have ported over minions from 4e (Matt Colville and I actually converged on this, I had been doing it since I switched to 5e and didn't find his video on it for years) and a modified version of the coup de grace rules. Minions are monsters with full stats and attacks but they die in a single hit, no matter how much damage they were dealt. For the modified coup de grace, if a player character deals half or more of a monsters HP in a single hit, even during normal combat, that monster dies immediately. Anything that gets the monsters off the field before they get boring really, since it allows me to throw out large waves of enemies that only take a few minutes to fight since many of them go down in one hit. I run a fairly heroic game of d&d so letting the players plow through enemies helps create the vibe I want during the game.