You did not mention whether you have beaten the Elite 4.
However, Mount Ember is small compared to the Seafoam Islands.
--FIRST VISIT TO MOUNT EMBER--
First of all, when you defeat Blaine, the leader in Cinnabar Island, Bill takes you directly to One Island's Poke Center. There you will meet with Celio, who is a technician trying to initiate the connection in computers through the islands. At this point you can get out of the Poke Center and head to the shore (right) and go north to Kindle Road. You surf past the Hot Springs (Ember Spa) to get to Mount Ember. At this point, you need surf, strength and rock smash to get to Moltres nest. You get rock smash from a guy inside the spa, talk to everyone there!
However, you have another mission to do here before you can head back to Kanto, and that is to go to Two Island to give a Meteorite to the Game Corner owner.
You will find however that he is too distraught by the fact that his daughter is missing, and you guessed it: you need to rescue her.
Head to Three Island and defeat all bikers, which is NO easy task. And head north west, pass the Bonding Bridge. There is a beach there but you must keep walking west until you reach the Berry Forest. When you are there, search through it and surf through a small lake to a place toward the west (inaccessible otherwise). Lostelle is a cute little girl who is frightened by a mean Hypno. Catch or faint that sucker.
You will be immediately transported to Island Two, then you can offer the Meteorite to Lostelle's dad. He will give you a Moon Stone in return. Now head back to Island One, to have Bill take you back to Kanto. You can come back to islands One through Three for now. And revisit Mount Ember.
The second cave in Mount Ember is NOT accessible yet, you must beat the Elite 4 now and return to it later.
--SECOND VISIT TO MOUNT EMBER--
After beating the Elite 4, you will be greeted in front of your home by Prof. Oak. You need to have caught 60 pokemon species, if you haven't; get hunting and go see Prof. Oak in his lab soon. Once you receive the National Dex from Prof. Oak you should head back to Island One. Have a chat with Celio, who will tell you he is attempting connection with the Hoenn region. That means you will be able to trade with Pokemon Ruby and Sapphire (and the consecutive games up to Platinum). He tells you he needs gemstones to make his machine work. Start by heading to Mount Ember. Now remember the Rockets you found conversing there earlier? Well, they have dug a cave into the volcano. What people do for money!
Anyway, you must battle them and the cave is your to explore. Keep going in the small cave which is not too big, and get the Ruby. This covers Mount Ember COMPLETELY. You can go out to the grass on Leaf Green and try catching a Magmar, which happens to be SUPER rare. But as far as completing tasks here this is it, nothing else to do here any more.