Did some more training tonight. I returned to Resort Gorgeous and rematched the trainers there several more times each. My Hitmonlee and Mr. Mime both grew to level 68 from these battles, and my Raichu grew a level as well. Then, I was bored with training in that area so I returned to Five Island and sailed back to Vermilion City. I went to several of the routes surrounding the Vermilion area to rematch some trainers I hadn't battled in a while. Raichu reached level 68 pretty quickly and then my Snorlax was able to grow to that level too.
Next time I do more training, I'm going to get Omastar and Venusaur to lv. 68 as well. But before I do that, there's something else I want to get out of the way first.