Hey, pretty fresh in terms of DMing and DND in general (been running this game for a little under a year but haven't had many sessions due to schedule conflicts).

My current party is an 8 person party, doing a original campaign and I'm wondering if anyone has any of their own large-numbered-party experiences they care to share or advice they might have for dealing with such a large quantity of people.

Unfortunately I don't have anything to add but empathy. My party grew to 6 and I'm experiencing slow combat. :-/

I run large parties as often as possible. I'm also one of the few folks I've known of online that don't think the best piece of advise for a party of more than 5 is "make your party smaller."

Here are some things to think about:

• Player involvement: Most folks are less interested in things that aren't directly affecting them, so it's natural that when it isn't a players time to be doing something with the game that they lose focus to some degree. That's not necessarily a problem, as most folks that lose focus like this can get it back quickly enough not to be a problem. It's only a problem worth booting a player from the group over if they just can't stay focused and just can't get back to focusing on the game when it is needed of them. Giving players more tasks related to the game to do can help keep them focused, and it can also speed up the general speed of the game by sharing what is typically the DM-workload among the group - have one player be in charge of tracking initiative in combat; another responsible for taking notes of important events and NPCs; get somebody interested in making maps for the party as they go about their adventure so if anyone needs to know where something is they can be asking the mapper, rather than the DM; you can also have players assist in setting up whatever battlemaps you might use, or even keeping track of how much damage monsters have taken. Anything that keeps them more constantly doing game stuff will help them stay focused on doing game stuff, rather than having the table devolve into a herd of cats that need to be wrangled when it is their turn.
• DM preparedness: I don't mean preparing at the adventure or campaign level - I mean preparing for the session so that the downtime during the session for things like book referencing, encounter set up, and the like are as minimal as possible. This starts with making good notes for yourself, like putting your campaign into this website and making sure to hotlink everything that you can so you just hover over it rather than flip pages in a physical book to get the relevant info, but extends into activities like pre-selecting battle maps or dungeon tiles and grouping those together with the minis that you'll use for an encounter. I have used ziplock backs in the past, labeled in marker so I could say "Hey [player], grab the bag labelled B9 and arrange it on the table, please." with a little scrap of paper showing the way the dungeon tiles and minis stored in the bag are meant to be laid out. I've also used various boxes (old box set boxes, board game boxes, things like that) for the same purpose.
• Using the right tool for the job: This ties into the above point, but is worth having it's own bullet point. Don't use minis and maps if the encounter is going to take less time than the set-up of those things - that's when it is best to run the encounter via "theatre of the mind". Save the use of maps and minis, which slow the game down as you set them up, for when they speed up playing out the encounter because there would be constant questions about the environment or position of the opposition due to their being too much information for everyone to keep straight in their heads.
• Player awareness and buy-in: A significant factor that often can go overlooked is just talking to the players. Saying "We've got a big table, and it'd work best if everyone did their part to keep things flowing." is probably all it will take to get the players to be aware, and their own fun being pointed out as being within their control is enough to get buy-in, but when these things go unsaid they often go unthought in my experience despite how obvious they seem.
• Other tricks: this is also a situation in which it is beneficial for every player to be trimming seconds off their turns by rolling color-matched attack and damage dice if they have multiple attacks, and by being ready to take and resolve their actions as soon as it is their turn, rather than not thinking of what they will do until their turn has come along. Which is easier if they each have the ability to reference whatever rules they might need to know (such as by sharing content on D&D Beyond with a master tier subscription).

I mean, I do have experience here, but this is such a broad question, that I don't really know how to begin.

You can start by reading my response to a similar question here.

Aside from that, it's just about recognizing that you have a large group and streamlining as much as possible. Know your stuff, do not add delays at the table yourself, and try to manage delays at the table coming from anywhere else. Be a bit of a phallus if you have to, but try to keep it moving. Try to limit the number of transitions. Try to plan out/schedule your pit-stops, so that your players don't suffer from mental fatigue.
If you have a player who is constantly derailing your campaign, you can put him in-charge of the map. This can streamline combat even more, and also give that player something to do to keep them interested/engaged.Utilizing inspiration to reward "paying attention" seems a little 4th grade, but it isn't a terrible idea.