Vermillion City!

Once you finally reach Vermillion City, visit all the buildings. You should end up with a bike voucher and an old rod. Use the old rod in the nearest water and capture magikarp. Train this pokemon (even though it's weak) so that it evolves into Gyarados (one of the strongest pokemon in the game!).

After that's done, go back to cerulean city and go to the bike shop. Give the man there the bike voucher and you'll get a bike. Now go back to Vermillion city.

Go into the S.S.Anne (docked nearby) and explore. It's better to face every trainer in the ship so that you get lots of exp. NOW IS THE BEST TIME TO TRAIN MAGIKARP! When you do beat every trainer including your rival, go to the captain's room and talk to him. He'll give you the CUT HM. Teach this to a plant pokemon. Now exit the ship. The ship will now leave.

Proceed to the vermillion gym and use a pokemon's cut to cut the shrub blocking your way into the gym. You can now face the Vemillion gym leader, Lieutenant Surge! He and everyone in his gym uses Electric pokemon. To face him, a ground pokemon (like geodude or diglett) would be super effective!!! But, to face him, you have to crack the code that opens the electric switches by pressing the 'A' button in front of a trash can. When you get one switch in a trash can, one of the cans next to it has the second electric switch. It took me about 15minutes to get it!!!

After you beat Lt. Surge, you get the thunderbadge.

The next step is to go to a computer and get your pokedex evaluated in prof. Oak's pc. If you have at least 10 pokemon by now, go through diglett's cave and when you reach the other side, get all the item balls lying around and then cut the shrub that's blocking your way into the next yard and go to that yard.

Go up until you reach the house with the aide. Talk to the aide(with glasses) and get the flash hm. Teach it to an electric pokemon. Go back through diglett's cave to vermillion city.

Now you have to go up toward cerulean city but go to the right hand side and cut the bush blocking the way. Follow the path until you get to the rock cave.