RBY Mewtwo is the single most powerful Pokemon in any generation. Nothing even comes close to the raw destructive power of this thing. That's all there is to be said about the joy that enters the heart of a 6-year-old when they capture the Uber and can truly call themselves a Pokemon Master.
Mewtwo is absolutely the most ferocious Pokemon that exists in RBY. Amnesia increases its already ridiculously high Special to incredible extents, allowing Mewtwo to wreak devastation on just about anything. After one or two Amnesias, Mewtwo is set to sweep: Psychic will be its main move and rends almost everything that doesn't resist it to shreds, while Blizzard hits the Psychic Pokemon such as Exeggutor that would otherwise have a chance against Mewtwo. Ice Beam is a more reliable option for Mewtwo and is the basic power versus accuracy argument. Blizzard does more damage per turn, factoring in accuracy, but with its low PP, it's not as reliable of a move to go for the freeze against enemy Mewtwo. Thunderbolt is an option over Psychic, as games in which Mewtwo are allowed are centered around countering it, meaning one will see even more Starmie and Slowbro, as they resist Psychic and the Ice-type move. Psychic is still the generally better choice, however. Recover heals Mewtwo quickly, but leaves it open to status such as paralysis; that's where Rest comes in and wipes any status off. Essentially, this Mewtwo set is the most powerful thing in Pokemon history.
This set eschews the massive power that Amnesia affords for more versatility. Psychic and Blizzard work the same way as in the previous set. The new additions are Thunderbolt and Submission: Thunderbolt smashes Slowbro and Starmie, which resist both Psychic and Ice, while Submission breaks down Chansey.
One should not forget that Mewtwo also has a base 110 Attack, which to put into perspective, is identical to that of Snorlax and more than that of Tauros. Submission destroys Normal-type Pokemon and Hyper Beam is a massively powerful attack to use against lower health Pokemon. Psychic is still here for reliability against many Pokemon; wasting Mewtwo's most powerful attack and that magnificent base 154 Special would be a shame.
Mewtwo can also be defensive. Barrier patches up Mewtwo's one "weak" spot, its base 90 Defense, and makes fainting Mewtwo incredibly difficult outside of a critical hit. Thunder Wave is always helpful to spread paralysis, one of the most powerful conditions in RBY. Recover is an obvious choice to prolong Mewtwo's survival even more.
Mewtwo does have other options, but most of them aren't worth it. Counter can surprise a Normal-type trying to take advantage of its "weak" Defense. Fire Blast is an alternate special attack and can burn quite often. Reflect ups Mewtwo's Defense. Selfdestruct is funny, and should generally go over Recover if you use it.
Substitute is decent in link battles, but it's godly in Stadium because it blocks status changes; paired with Recover or Rest, Psychic, and Amnesia, it makes Mewtwo even more unstoppable than it already is. As you are most likely not playing Stadium, and the other options stated here aren't particularly great compared to the movesets above, it's better to stick with those sets, as they'll work, and very well.
Checks and Counters
Nope. Nothing can fully counter a Mewtwo; some can make it think twice, though. Chansey is probably the best counter. Light Screen Chansey, more specifically, can stall it and aim for a freeze. However, Mewtwo has a high critical hit rate and Amnesia on his side. What's more, it could backfire on you; Mewtwo can set up easily against Chansey, and will get to 999 Special after two Amnesias. If it gets past Chansey, you can say goodbye to your whole team. Mewtwo will also be aiming for the freeze on your Chansey with Ice Beam / Blizzard, or just spamming Psychic for a Special fall.
The Psychic / Blizzard model is stopped cold, as far as resistances go, by Slowbro, and, to an extent, Starmie, but they can't touch Mewtwo either. Also, watch out for Snorlax with Selfdestruct.
If you ever face a Mewtwo, paralyze it at all costs, unless you have a Mewtwo, Slowbro, or Chansey of your own, and want to play dice with their Mewtwo by going for a freeze. This cannot be stressed enough—a Mewtwo that is not paralyzed will tear apart your whole team. You CAN stop a paralyzed one with some luck and one sacrifice if your opponent is stupid enough not to switch out. A fresh Mewtwo will destroy every single one of your Pokemon after setting up, and you won't be able to do anything about it. And, well, if Mewtwo has Rest and your opponent is skilled... good luck.
Physical blows are where Mewtwo is most vulnerable (although it's not weak on that side, with defensive capabilities about halfway between Tauros and Slowbro), but with an Amnesia, Mewtwo can devastate all of the physical attackers that it is weak to.