Assuming they are starting as level 1 characters, prepping for the very first session should involve close scrutiny of the Tier 1 information (Levels 1 - 4). While there's no harm in reading ahead as an FYI, you want to be clear on the material that is likely to involve them in their immediate next session.
They should not encounter any of the detail of the Tier 2 material beyond lore and story as you mentioned; it's good to read through it all and see how the story plays out, but the gameplay itself will suffer if you mix in too much encounter material from later tiers.
As a new DM you will be astonished at how little progress your players make through an adventure in any one 3 hour session; They may only get into a single area and end up staying there all night. 4 encounters is normal, 6 feels like a lot; sometimes there may be none, when characters are restig in a tavern, restocking/resupplying or just carefully investigating non-combat areas.
The advantage of the prewritten modules, and especially linear dungeon areas, is that you can usually determine what the next 4-5 encounters are most likely to be, and ensure you're prepped for those (mainly, familiarizing yourself with monster capabilities, their best attacks/combos, their escape routes, and their motivations) as well as traps, items of interest, and loot/treasure info. If you are using wandering monster encounters, prep the same for those - basic monster stats, capabilities, motivation - why are they wandering, after all?- plus escape routes, and treasure, if any.
When creating and prepping your own material, this is still a good rule of thumb - identify and prep the next 4-5 most likely encounters and you'll be in good shape. The advantage with homebrew is you are always off script, so you're more free to come up with your own filler when characters immediately start breaking through a wall and into an area of the dungeon they weren't supposed to get to for six more weeks.
The downside with homebrew is that you abandon a lot of material, but you can be more effective there too by only prepping for 4-5 encounters per session so you don't feel so bad about losing weeks/months of prep on an elaborate plot that they are now totally uninvolved in, in favor of some other thing you threw in on a whim in session 1 (that's me right now, in Session 17, where the players diverged in the very first episode based on my idea of having them find a baby instead of treasure in a basket).