Heard this asked so many times so im going to run through it all. You will need a pokemon with?strength and surf?for this.
Step 1:?Defeat the Elite Four
Step 2:?Fly or boat (via Vermillion) to island one.
Step 3:?Go inside the pokecenter and talk to Celio by the big machine
Step 4:?He will mention something about a Ruby, at this point you need to go to MT. Ember (also on island one)
Step 5:?Defeat the team rocket guys off to the right of the entrance to MT. Ember
Step 6:?Enter the cave and follow the path through
Step 7:?Complete the first 2 rock puzzles, at the split road take the one to the left and follow through until the Ruby
Step 8:?The pathway to the right is a faster way out the cave
Step 9:?Fly (or walk) back to Celio with the Ruby in hand. He will upgrade your ferry pass to allow full access to the other islands.
As per a CM request here is another guide with directions on how to get the Sapphire.
You will need a pokemon with?Surf?and one that can learn Waterfall?for step 1. After step one you must have a pokemon that knowsCut?and?Strength
Step 1:?You must defeat the Rocket Grunts inside Icefall cave (island 4). To do this you must first complete the Ice Puzzle and obtain HM Waterfall at the end of the cave. Once obtained return to the front of the cave and use Waterfall on the Waterfall (duh)
Step 1.5:?Head to island 5
Step 2:?Go to Five island Meadow and defeat the Rockets outside the Warehouse to obtain the first password
Step 3:?Fly/Boat to island 6
Step 4:?Head out East from the Pokecenter and Bike/Walk south
Step 5:?Follow the path until you reach Ruin Valley
Step 6:?At Ruin Valley take left path and follow it through, the two paths that require strength lead to HP UP and a Full Restore
Step 7:?After following the left path you should reach a small closed cave. The door should have Braille on it
Step 8:?Use Cut on the door (by selecting your pokemon, going to moves and clicking it)
Step 9:?Enter the cave and fall through the first hole
Step 10:?Follow the remaining holes in the following order.
Step 11:?Click the Sapphire, watch some noobish Scientist steal it. Take the stairs back up and leave the cave
Step 12:?Head to island 5
Step 13:?Cut through the Five island Meadow and go to the front door of the warehouse and hit A on it,
Step 14:?Enter the Warehouse, follow the floor paths through. Upon reaching the end of the floor paths a Rocket Admin will engage you in a battle
Step 15:?After defeating the Admin head through and destroy that pesky scientist.
Step 16:?The scientist will give you the Sapphire, with that return to Celio on island 1 and hand it to him.
With those done you now have access to Cerulean Cave and numerous high level pokemon. Enjoy