For training Pikachu, there are several possibilities. There are some trainers east of Vermillion city. Then there's the SS Anne, which might have trainers with suitable pokemon for training Pikachu.
Once you get onto route 12/13/14, there are some Bird pokemon trainers. Also, there's a house to get a super rod, which will allow you to catch some good water pokemon like Gyarados to train on.
There are more places, just go through the cities in the order in which they are accessible, and use Pikachu for any Bird/Water pokemon you encounter, either wild or trainer owned.