[RPG] What sort of cheat sheets for a new DM lead to a better flowing 5e game of Dungeons and Dragons

This will unfortunately require a bit more work than you probably want to do when running the starter, but it's doable. You need a copy of this document. Take a gander at the second to last chapter and there you will find the encounter construction rules.

Basically what you need to do is find the encounter construction rules, reset the XP budgets to be for 6 players, and then rebuild each encounter with the appropriate number of monsters.

I would note that this is completely optional. You can run this straight up, let your party mow through it, and simply chalk it up as a learning experience.

One last suggestion if you don't want to change anything at all, is to adjust monster HP and damage. This is something I've experimented a bit with in my games with fewer people, but it's something that is definitely worth trying as monster HP and Damage are significant scale factors in 5e. Each monster has their average HP and damage listed. use a midpoint between their average and their high.

Is playing through The Mines Of Phandelver (hope I got that right) and looking things up in the rules while doing so an option?

The Lost Mines of Phandelver starts out relatively simple, specifically for beginners, so don't worry.

My group had several new players. Here's what we did, and it worked well.

Prior to the First Session

1. A couple players read the rules pretty thoroughly (...for fun).
2. Everyone comfortable with making their own characters did so; everyone else chose prebuilt characters from the starter set box.
3. The DM read the first bits of the starter set adventure to get a feel for how to run the game and the first few encounters.
4. I wrote up a brief introductory email with an overview of the rules (I'm not sure everyone actually read it, which is fine).
5. I also printed up a few copies of a cheat sheet we made up, which we used for the first couple sessions.

During Sessions

When rule questions came up, the DM made snap decisions that seemed fair and interesting. We looked up the actual rules later. This didn't break up the flow of the game and worked just fine.

It doesn't matter if you don't get all the rules right, so long as everyone is having fun and things don't grind to a halt.