The Egg Corridor
When you start, head right and be sure to never touch the creature along the bottom, as it will kill you instantly. Be sure to kill as many enemies as you can, because a level three Polar Star is very useful here. Eventually you'll get to a marginally tall pillar. Jump over it to view Sue getting attacked and kidnapped by a monster. After you regain control, continue along your way.
Around the half-way point you will encounter a door. Going inside the door leads you to Cthulhu's Abode, where there is a save point, health station and a strange man. Save if you want, then continue along your way. When you reach the area around the 6th egg, wait for Basil to pass, then drop down one of the holes and run and jump into the chamber containing Egg No. 6. Jump up towards the center of the egg and you will be brought to a special room with a treasure chest. Grab the chest to get the I.D. Card, then leave the area the same way you came in.
Eventually you will come upon another door. This one leads to the Egg Observation Room, where you can save. Also, take the elevator to the upper floor to get a new weapon, the Missile Launcher. Try to level it up as fast as you can for it is extremely useful.
Continuing along your way, you will run into a barrier. No problem, just drop down into the lower corridor with Basil and run to Egg No. 1 like you did with Egg No. 6. Don't bother with egg No. 2, it's just a place to hide from Basil. Inside Egg No. 1 is a computer. Use the I.D. Card on the computer to lower the barrier and leave the same way you came in. Hop over the now gone barrier and you will enter a boss fight with Igor, the monster that kidnapped Sue (strategy here). After Igor is gone, enter the left door to enter Egg No. 00. In Egg No. 00, walk over and talk to Sue. A dialog will occur then she will leave. After she leaves, follow her outside.
Exiting the Egg Corridor
The right door can now be opened. Inside it is a save point and a health station. Use them if you want, then start to head back to the teleporter. Be careful however, because there are now many more flying enemies in the Egg Corridor. Most of them are easy and can be killed in one hit, but watch out for the larger, stronger Basu. Play it safe and smart and save or refill your health at all the save stations along the way.
Once you get to the teleporter, warp back to Arthur's house. You can revisit the Egg Corridor any time you want.
After viewing the scene between Sue and her brother Kazuma, go outside Arthur's House. Talk to Sue and she will ask you to save her brother, since she is now imprisoned. Save, then go back to the teleporter and warp to Grasstown.