The Omega Ruins are where you will find some of the most difficult fiends to fight in the game including Great Malboros and Tonberrys. Make sure you prepare yourself by equipping at least one of your characters with armor that can prevent or ward Petrification (either the 
Stone Ward or Stoneproof auto-ability).
You should also make sure that you have a weapon with the First Strike auto-ability. Read the section below for more information on how to obtain it and then read the information window below for more information on how to defeat a Great Malboro:
Great Malboros will always ambush you when they encounter them. The best way to defeat them is to have one character equipped with the First Strike auto-ability. You can then use 
Provoke on the Great Malboro which will prevent it from using “Bad Breath” as its first attack.
You can find the Omega Ruins by unlocking them from the airship (along with a number of other locations). Check out the Airship Passwords and Hidden Coordinates section for a list of all of the secret locations.
Note that, similar to the Sea of Sorrow, you have to walk through the Omega Ruins in order to fill in the mini-map (similar to a fog of war). A rough drawing of the map of the Omega Ruins below:
There are four clusters of treasure chests located throughout the Omega Ruins. Each of the chests will either be an actual treasure or you will be thrown into a battle against a Mimic.
As soon as you fight the Mimic the other treasure chests will disappear. The list of available treasures are included below, however you may not be able to obtain them if you run into a Mimic:
1) 
Lv. 4 Key Sphere
2) 
Defending Bracer
3) 
Turnover
4) Lv. 3 Key Sphere (x2)
5) 
Defending Armlet
6) 
Friend Sphere (x2)
7) Lv. 4 Key Sphere
8) 
Phantom Ring
9) 
Cactuar Wizard
10) 
Warmonger
11) Teleport Sphere (x2)
12) Warp Sphere x99
The Final Fantasy Wikia page has a very good guide on how to obtain the last treasure of Warp Spheres (x99). It could be argued that they are not worth the amount of effort that it takes to obtain them but you will have to decide on how much time you would like to invest in obtaining these rare items.
Travel to the Glyph that is furthest north on the map above. Pick up the 
Al Bhed Primer (Vol. XXVI) on the ground along the way. It is located near the batch of treasure chests.Touch the treasure chest when you reach that area of the map. Then head back to the southern most Glyph and touch that one to cause the platform to the south to raise.
Other parts of the platform will raise as you work your way towards the end. There is a treasure chest at the end that contains a Teleport Sphere.
Navigate your way to the upper right area of the map to proceed to the next level of the dungeon. This is easier to do if you have armor with the No Encounters ability equipped.
As soon as you enter the small room you will be thrown into battle against Ultima Weapon.
HP: 99,999
Ultima Weapon should be, and likely will be, an incredibly easy boss to defeat. Spend some time leveling your characters up (near the entrance with the Save Sphere so you can refill your HP and MP) and this boss will be a breeze. It only has 99,999 HP.
You can also bribe Ultima Weapon with 1,400,000 gil in order to obtain 99 
Pendulums which can be used to add the Master Thief ability to an armor (which causes you to always steal the most rare item). Definitely worth it if you can afford the gil. If not, you could level up in Omega Ruins, selling armor that you receive, using the Pilfer Gil ability and obtaining enough gil to bribe Ultima Weapon.
Note: This amount is increased to up to 2,000,000 gil in the international and HD Remaster editions. Just continue to Bribe 1 gil each time after this if it misses as once you reach the proper threshold amount of gil each Bribe attempt has the same chance of working.
Steal:
Door to Tomorrow (x10) (common)
Door to Tomorrow (x20) (common)
Drops:
Lv. 3 Key Sphere (x3)
Take the platform/elevator up to the next level to continue.
There are two pathways that you can take. The pathway to the left leads to a Friend Sphere after two encounters and the pathway to the right leads to Omega Weapon. You will be thrown into a fight automatically regardless of whether you have the No Encounters ability equipped each time you enter a new platform.
The most difficult fights are definitely against Great Malboros. Each time you fight them they will automatically ambush you and will use Bad Breath as their first attack. Make sure that you are prepared for this by having a weapon that has the First Strike auto-ability.
Save up your Overdrives for the upcoming boss battle. You will be thrust into battle once you enter the glowing area at the end of the pathway.
HP: 999,999
Omega Weapon is an entirely different fight depending on what version of the game you have. In the original version of the game it has 99,999 HP however in the international and HD Remaster versions it has 999,999 HP.
This can be a relatively lengthy battle so you should start off by casting 
Hastega, Shell and Protect on each of your characters. Alternatively, you can use Rikku’s Mix ability to cast either Super Might G or Hyper Mighty G on your party.
Use all of your Overdrives and use 
Quick Hit to get as many attacks in as you can between Omega Weapon’s hits.
Make sure you take the opportunity to steal Gambler’s Spirit items during the fight. If you are having trouble with this fight it is advised that you spend more time leveling your characters in the Omega Ruins or to complete some of the side quests available, namely the Celestial Weapon side quests, as breaking the damage limit can make this fight relatively easy.
Make sure to have Kimarhi use Lancet on it so that he can learn Nova, his strongest Overdrive.
Steal:
Gambler’s Spirit (x30)
Drops:
Lv. 4 Key Sphere (x3)
The game automatically warps you back to the entrance of the Omega Ruins as this dungeon is now complete.
