Red and Blue have the simplest evolution system in the series. Pokemon evolve by reaching a specific level, using one of five evolution stones, or trading with another player. There are no friendship requirements, no held items, and no time-of-day conditions. If a Pokemon doesn't evolve by level or stone, it evolves by trade. That's it.
Three methods total. Level-up (most Pokemon evolve at a set level), evolution stones (Fire, Water, Thunder, Moon, and Leaf Stones), and trading (four Pokemon evolve when traded: Kadabra, Machoke, Graveler, Haunter). No friendship, no held items, no time of day. Gen 1 keeps it simple.
Celadon Department Store on the fourth floor sells Fire, Water, Thunder, and Leaf Stones. Moon Stones are found in Mt. Moon (two of them), the Pokemon Mansion on Cinnabar Island, and the Rocket Hideout. You can't buy Moon Stones. One playthrough gives you just enough for your team if you plan carefully.
No. Evolution methods are identical across both versions. The only difference is which Pokemon you can catch: Sandshrew is Blue-exclusive, Ekans is Red-exclusive, and so on. But once you have the Pokemon, it evolves the same way regardless of version.
Sometimes. Stone evolutions stop learning moves in Gen 1. If you evolve Pikachu into Raichu immediately, Raichu won't learn any new moves by level-up. Same with Vulpix, Growlithe, and the others. Let them learn the moves you want first, then use the stone. Level-up evolutions don't have this problem.