[X] Seek out a Chansey - Chanseys are located mainly within Cerulean Cave, a locale where you are most definitely ill-qualified for. However, there are a few branches of the cave which are accessible to you.
[X] Challenge Luna - A known trainer with a decently trained Growlithe, she would make for a good sparring partner while you're here. And maybe let you talk in the future.
[X] Challenge an Experienced Trainer - The other option to seek out a Pokemon battle would be to go to the Cerulean Gym and challenge one of the attendants there. You would likely even win.
[X] Call Willow - The trainer you recently defeated, she seems fun if exhausting. She just raced through the Viridian Forest and plans to challenge the Pewter City gym.
[X] Explore some Caves - Now that you have learned the basics of caving, taking on more challenging spelunking expeditions is viable. And there are a fair few options nearby.
[X] Learn to Swim - While you do know how to swim, there is a notable difference between swimming for fun and scuba diving. And an even greater difference when it comes to exploring underwater caves. Learning more advanced concepts would be fun.