Gaining Levels
Pokemon gain levels by earning experience from successfully winning battles against other Pokemon trainers and wild Pokemon. After a pokemon's experience bar is full, that pokemon gains a new level. Pokemon often learn new moves every four to seven levels. For example, Bulbasaur learns Leech Seed at Level 7, Vine Whip at Level 13 and Solar Beam at Level 48. The amount of moves your Pokemon can learn as they level depends on what Pokemon you use. Most Pokemon don't learn additional abilities until they reach a certain level, which for many is a level in the 50s. Exceptions to this are Mewtwo, who continues learning moves through Level 81, and Starmie, who learns no moves at all.
Machines
In addition to learning moves by leveling, pokemon can also learn moves from machines. Two types of machines exist in "Pokemon Blue": Technical Machines (TMs) and Hidden Machines (HMs). HMs teach pokemon moves that are useful inside and outside of battle. For example, Cut allows pokemon to cut down small bushes, while Surf lets you ride a pokemon in a body of water. HMs can be used more than once. Alternatively, TMs can only be used once to teach a pokemon a move they wouldn't otherwise learn, such as Toxic or Bubblebeam. TMs and HMs can be used on pokemon at any level, as long as they are of the right type to learn the move.
Pokemon Move Limit
All Pokemon are limited to knowing four moves at once. If a Pokemon wants to learn a new move after it knows four moves, it must forget an old move. By forgetting a move, the Pokemon frees up a move slot so it can learn the new move. In "Pokemon Blue," it's not possible for a Pokemon to recover a forgotten move. Carefully select which moves to forget and to keep, as you will often not have the chance to re-learn a move after forgetting it.
Evolving
When a Pokemon evolves into a new form, such as Bulbasaur to Ivysaur, it increases its statistics, such as receiving higher attack power or more health. However, the evolution comes with a price -- evolved Pokemon learn moves at a slower rate than non-evolved pokemon. For example, Bulbasaur learns Poison Powder at Level 20 and Razor Leaf at Level 27. Ivysaur, Bulbasaur's evolved form, learns Poison Powder at Level 22 and Razor Leaf at Level 30. One tactic players use is to not evolve a Pokemon until it learns nearly all of its moves; this way, Pokemon don't experience a level delay in learning new moves.