After Mt. Moon, Route #4 will seem like a walk in the park. There are no trainers to challenge you and only one area where you can run into wild Pokémon. Be sure to collect Whirlwind (TM04) before you jump the hedge to reach Cerulean City. After you jump it, you won't be able to jump back over it to backtrack...
Next stop: Cerulean City
There's nothing particularly interesting on Route #4. All you'll find at this point is an item ball with TM04 in it, which teaches a Pokémon the Whirlwind skill. However, it can only end wild Pokémon battles if it hits, and using them in a trainer or link battle will do nothing unlike the following generations.
There is a hidden Great Ball (not designated on the map), a powered-up Poké Ball. Look on the first plateau of sorts (formed by the ledges which one can jump on all sides); it will be around the middle of the highest level.
Be careful: When you jump down the wall to the southeast (and you'll have to in order to progress), you won't be able to go back!