Go to the next gym leader, Chuck, who is in Cianwood City. You can get to Cianwood by first going to Olivine City (which is West of Ecruteak City) then going through Olivine City, turn West again and you will be at a beach route. Then teach one of your Pokemon HM surf (which you get from the kimono girls in the Ecruteak Dance Theatre after beating the Team Rocket Grunt) and press A when you face the sea and you can surf. Once you get through the sea route, you will eventually arrive at Cianwood city, where the next gym will be. (Also on the way to Cianwood you will see a cave you can't get to; those are the Whirl Islands, if you are playing Heartgold, the islands won't be much use to you)
After you beat Chuck, you will receive HM Fly from his wife, which lets you fly anywhere you've been to. Also, before you leave do two things:
Go North in Cianwood, you will find Suicune and Euscine again, and Euscine will battle you.
Go to the Cianwood City Pharmacy and get some SecretPotion, you will need it later in Olivine City, at the lighthouse, before you fight the sixth gym.
Tell me if you need any more help.
Hope this helped.