Okay, here's what you do. First, you snag yourself a water type (I recommend Vaporeon if you didn't start with Totodile: just go to Bill's house west of the Pokemon Center in Goldenrod City and then you can get an Eevee from him, then you have to get a Water Stone in order to evolve it [you can get Water Stones for 2500 Athlete Points if you're lucky], otherwise you can have a Sentret or snag a random water type from having the Good Rod [received from a Fisherman in Olivine]), then head on over to Ecruteak City.
There is a Dance Theatre there (it's slightly northwest of the Pokemon Center if I remember correctly) and in there you'll see a lone Team Rocket Grunt harrassing a Kimono Girl (must... resist... urge to... make... Mother 3 reference...). Go on stage and speak to the Team Rocket Grunt. He'll challenge you to a battle and send out a Koffing. If you survived the Rocket Grunts in the Slowpoke Well, you'll survive this one easily. Especially since he's only got one Koffing and it's level 11 (I believe). Just nuke it with Togepi/Togetic/Togekiss (if you have one) Extrasensory or fry it with other special based attacks (IE Water Gun, Ember, and the like) and you'll be fine. For your efforts, you'll receive HM03 Surf from her which will allow you to go to Cianwood City.
Once you've done that, go west of Olivine and press A to examine the water which will allow you to Surf across the water. Head along the rather linear route (ignore any caves or Whirlpools you see on the way: you can't do anything about those at your point in the game anyway) and you'll find yourself in Cianwood City. There's a Pharmacy in one of the houses (which is where you get the Secret Potion, the Medicine that Ampharos needs to live). However, if I recall correctly, you need the Badge from Cianwood to get it (Cianwood Gym specializes in Fighting types so bring in a Psychic user [Togetic and Espeon (you get Espeon by leveling up an Eevee with maximum friendliness during the Day) are amazing for that] or a Flying type [the Pidgey and Spearow families are pretty good for that]).
Hopefully that wasn't too long winded, and I hope I helped you.