Where is team magma's hideout and what do you do at lillycove?

1st answer:
Go first at Mt. Pyre and and don't go to the stairs, go to the other exit from the first floor. Go to the top of the Mt. Pyre and the Team Aqua is there, and when you reach Archie, he will steal the Red Orb and the old lady will give you the Magma Emblem. She says that the Team Magma is there first and they leave the Magma Emblem.
The Magma Hideout is in Jagged Pass, so may take the Cable car or jump from the rocks of the Jagged Pass using an Acro Bike. Go up/down, and an earthquake trembled. Go inside and find Magma Leader Maxie.

2nd answer
If you're at Lilycove City, you can go shopping or enter the Pokemon Contest. That's it.

Hope this helps and Good Luck!

Well, again, Jordi beats me again, but I do believe your question is about getting Archie out of Lilycove. All you need to do is beat Maxie, then fly/walk/run/bike to Slateport. Go to the actual port at the northeast part of the city, where a large crowd is gathered. Capt. Stern will be ending a speech when team Aqua comes on the megaphone. They have stolen Capt. Stern's sub! Once that whole scene's over, head back to Lilycove, and surf off the northeast coast to the Aqua Hideout. You should be able to get past the guards, and navigate yourself to Team Aqua Admin at the end. Once you beat him, you should be able to get past the sea barricade and head for your next destination: Mossdeep.

A helpful tip: If you find yourself in a fancy room with four pokeballs in the corner, the left column should have a MASTERBALL and NUGGET.