You don't get the Super Rod (which is the only fishng pole in the game) until you beat the story, and you won't get HM03 (Surf) until you get to Twist Mountain.
You can get your first Water-Type Pokemon WITHOUT Surf and the Super Rod in Icirrus City. Just walk in the puddles. Check the second link for info on the Pokemon you can find.
For more info: Bulbapedia/ Twist Mountain
For info on Icirrus City : Bulbapedia/ Icirrus City