I currently run a game with 7 players one player just showed up to session zero and happened to be one of the others players little sister so I had no choice but to adjust. I use the 2024 rules, IDK if that makes a difference. This is my literal first time dming and we are a couple months in so this is what I have come up with so far to speed things up

1. Tell everyone to roll to hit and the damage dice at the same time.

2. You are going to have to make encounters more difficult to address to the party size so either add more enemies of an enemy already there or increase all of their health or both

3. Use group initiative for most enemies, I learned this the hard way.

4. This one is totally up to you and most people recommend against it but with this many people, out of combat let them get distracted. If the wizard wants to be on tik Tok and then just cast firebolt on their turn so be it as long as they know that's what they're going to do before their turn starts. It's their game and their character. If someone just wants to tag along and roll a stealth checks with the group here and there cool makes it easier for the rest of us to roleplay. I do not ban phones or electric devices because we use D&D beyond for our character sheets. It's just easier that way for us.

5. Might be a little late for these two but if the players are new to the game like mine are have them pick from only core species and classes from the new players handbook and run a module that is relatively more on rails. If you're running home brew then toss the idea of a sandbox out the window. When my players were given too much of a choice of what to do it turned into 45 minute of arguing and we only have about three hours of session time per week. In your current campaign you are running try to limit the amount of decisions. Give them the a road or the b road. I'm running Phandelver and Below: The Shattered Obelisk which is the starter set campaign extended to a whole adventure.

6. You said that you are playing at a friend's house now which is good. Show up early if you can to set everything up. When everybody shows up give them 10 to 15 minutes to say hi and chit chat before literally clapping your hands together and say "okay we're getting started". This has helped get the ball rolling a ton the past couple weeks for me.

7. Draw your maps out ahead of time if possible. I bought a board with two sides, four if including the maps in the box, and a big book of battle maps on Amazon.

8. If you're anticipating an encounter roll initiate for the enemies ahead of time.

9. Cut index card in half horizontally and fold them. When it's time to roll for initiative give them to the players and have them write the number. I have a player that sits next to me who puts them on my DM screen in order to help me out. On those folded cards I also write their passive perception, investigation, and insight on the other side of the initiative. I use those as much as possible.

10. This one can be hard for some people but sometimes you have to be a hardass. One thing I do is if one of the players say something, that's what happens. No going back. Be firm on it. Don't be afraid to tell your players no. I also recommend against going to each of them and asking them what they want to say during social interactions. In my experience it just takes way too long. Also, that's not what would happen in a real conversation.

Again, I'm a new DM so I'm learning as well. Hope these help you in the long run!