"Conservative Party HQ. OOOOOOH triggered?"
Whether you bust on in or take the stairs is up to you, but if you do take the stairs, take the lift back to the ground floor and read Turtles Paradise 2.
On floor 62, correctly guess the password on first try and you'll get an Elemental Materia. Basically, there are four library sections, each a different category. In each category all but one book will be appropriate for that category. So in the "Weapons" category there will be a single book NOT about weapons. The book will have a number (let's call it N) and a description. The Nth letter of the description is one of the letters of the passw... oh forget it. Just go to the first floor, save, and keep trying/reloading until you get the Materia.
On floor 63, you can collect an A Coupon, B Coupon and C Coupon. To get them all, go to the very top wall, open the door, open the two doors below and go into the leftmost room. Grab A Coupon. Climb into the air duct to get the other two. Now, you could exchange for a Star Pendant, Four Slots and All Materia. But don't do it. You can get more of these relatively easily in the game, but the coupons are gone forever! (Similar to the Battery situation).
On floor 67, be sure to grab the Poison Materia in the treasure chest before entering the lift to fight the boss and recruit Red XIII.
On floor 68 where you recruit Red XIII, grab the Enemy Skill Materia after the ol' boss battle there Ted. This IS a permanently missable, there are only four in the game. Make sure you get it!
On floor 68 again, I *think* it's actually possible to permanently miss Keycard 68. So make sure you get it off the guy to the right after the boss battle.
Not permanently missable, but great to have at this point. Run around on floor 68 until you battle a Soldier 3rd. Steal the Hardedge sword from 'em.
Also not a perma-missable, but great to have. Steal Carbon Bangles from Moth Slashers on this floor (those robots that are on top of spheres).
Alright, you can forget about missable items for a while now! Come back once you've exited Midgar.
After a little chat, choose a party and exit Midgar by just moving to the bottom of the screen.
Alright, before moving on you're going to want to roam around outside Midgar and have a few battles. You can learn enemy skill Matra Magic from enemies called "Custom Sweepers" here. These enemies are robots with guns for arms and cone-shaped feet. Equip someone with the Enemy Skill Mateia and stand around until a Custom Sweeper uses Matra Magic. This attack deals damage to all enemies and doesn't cost too much MP.
With regards to Enemy Skills in general, I tend to pick up good ones as I go through the game, but I wait until I've gotten all four Enemy Skill Materia before I go around learning them all. Don't worry, this guide will make sure you can have all four Enemy Skill Materia and every single enemy skill! So press on!
Okay, you may also want to steal an Atomic Scissors from a Custom Sweeper. It's not a permanently missable and can be purchased in shops later. But it's a pretty powerful weapon for Barret at this point in the game. So get one.