Pidgeot is easily one of the most underrated Pokémon in RBY. While it may seem like it can't do much, it can actually hold its own against a lot of OU Pokémon thanks to Mirror Move and its Normal STAB. Pidgeot works best as a supporting Pokémon, either setting up for other Pokémon on your team or sweeping weakened Pokémon. Because there are a lot of things Pidgeot can do, there is no single best set for it. Pidgeot is a decent Pokémon that can work extremely well in UU and can often do well in OU as well, if you use it right.
This moveset is quite effective, but you could use a variety of other moves like Sand-Attack or Fly on Pidgeot successfully. In general, Pidgeot should definitely have Mirror Move and either Hyper Beam or Double-Edge as a basic attacking move. Mirror Move is Pidgeot's signature attack and it helps makes up for its lackluster movepool by letting Pidgeot use its opponent's attacks against them. Using Mirror Move depends a lot on prediction, but Pidgeot's defenses are good enough so that it can take a variety of attacks to send back against the opponents. Double-Edge and Hyper Beam are good basic attacks and finishing moves. Agility and Substitute both work well with Mirror Move by letting you see whether the opposing Pokémon's attack is worth Mirror Moving, and Agility also helps it sweep. Mimic works extremely well with Pidgeot for the same reasons that it works with Zapdos; you can try to get Earthquake from Golem or Rhydon on the switch, as well as something like Thunder Wave from Starmie.
As already mentioned, Sand-Attack is an option for inducing switches, and Fly is a STAB option. Reflect can be considered for defensive purposes, but it's not like it'll do him any good against Rhydon or Golem. Swift, Quick Attack, and Sky Attack are all decent attacks that Pidgeot could benefit from, given the situation.
Checks and Counters
Golem and Rhydon both resist Pidgeot's moves while beating it with Rock Slide. Gengar cannot be hit by Pidgeot using Normal moves, and can easily beat it with Thunderbolt. Starmie can beat it with Thunderbolt or Blizzard, while at the same time recovering off damage it takes from Pidgeot. Electric-types will pretty obviously beat it with Thunderbolt as well. Cloyster has a huge Defense stat to absorb damage, and will be able to Blizzard Pidgeot to a quick KO.
In UU, Aerodactyl will beat Pidgeot, and Poliwrath will switch in to use Amnesia if it finds out you aren't using Fly. Omastar and Kabutops take little damage from Pidgeot and can Blizzard it. Likewise, Onix takes little damage from all of Pidgeot's moves and can use Rock Slide to be an effective counter.