In GSC the only way to tell is either by evolution (If Eevee or Golbat are happy enough, they'll evolve for example), or there's a woman in Goldenrod who will tell you how much your Pokémon like you. Though sadly I don't have a list of the phrases she uses depending on happiness.
Back in Gen I you only had Pikachu to worry about lol, and you could tell how happy it was simply by speaking to it.