Abra -
Top Tier
Availability: Early, before second gym. While hard to catch, Butterfree, Clefairy and Jigglypuff all learn Sleep Powder, and with RBY sleep mechanics, you can just keep putting Abra to sleep until you catch him so he doesn't teleport. Alternatively, you can wait until you reach Celadon to buy him for 180/120/230 (R/B/Y) coins.
Stats: Above Average
Movepool: Very limited, only gets STAB moves and utility moves. However it learns everything it needs by level 38 and can afford to run monotype offense.
Power: Having huge base stats in speed and special make Abra a huge threat to everything else. The strongest Psychic type in the game that isn't Mew or Mewtwo when it evolves, this gives it pure type advantage against everything. Confusion, Psybeam and Psychic are all really powerful in the game and hit everything that isn't a psychic type for neutral damage. It's worth noting that even if you can't evolve to get Alakazam, Kadabra's base Special is 125 and it's base Speed is 105, well above average. In addition, their speed means that they will both be dealing critical hits around 1/3rd of the time.
Type: No pokemon in RBY resists Psychic aside from other Psychics. In addition, Psychics resist Fighting and Poison, are immune to Ghost and are only weak to Bug which don't have any threatening moves in this game.
Match-ups:
Misty: If it's an Abra still it won't have any offensive moves. Kadabra/Alakazam can take Staryu and do some damage to Starmie, but it won't win unless its overlevelled or you get lucky with what moves it chooses.
Lt Surge: Kadabra makes short work of Voltorb and Pikachu. Raichu packs Thunderbolt which will hurt, but he shouldn't use it more than once on you depending on your level. Raichu can KO, so make sure to have a back-up waiting in the wings to finish the gym off.
Erika: If you used Kadabra/Alakazam to fight the rockets they/he should be more than strong enough to sweep the entire gym. You should OHKO everything in the gym with Psybeam, even Tangela who isn't weak to it. You can waste your Psychic TM on him at this point, but he learns it naturally in 10 levels anyway.
Koga: Way too easy with Psybeam. There are Jugglers in this gym who use Psychic-types, but none of them are a problem. Koga's entire team is weak to Psychic and nothing he has is as fast as Kadabra/Alakazam
Blaine: Will have Psychic by now and will have no trouble. Can take even Fire Blasts with its monster special but watch for Rapidash's Stomp or Arcanine's Take-Down.
Sabrina: Psychic is only resisted by other psychics, and Kadabra/Alakazam is the best of the bunch, so while the other psychics will resist your attacks, you will resist theirs in-turn. Watch out for Sabrina's Alakazam, for the same reasons that everyone else should watch out for yours. If you fought the gym leaders in this order though, you should be over-leveled though and have no trouble.
Giovanni: You are faster than all of his pokemon, and they all have terrible Special stats. You should sweep the gym no problem, just make sure to go all out as you can't take a single physical attack from anything. Your speed means you don't have to worry about Fissure or Horn Drill.
Lorelei: At this point you sweep her entire team and can take pretty much all special moves they can throw at you. Just don't go up against Jynx unless you have to, as it resists Psychic and can significantly wear out your sustainability.
Bruno: Sweeps. Of the 5 pokemon Bruno uses, the only 2 not weak to Psychic have terrible special stats and are incredibly slow. You should not have a problem.
Agatha: All her pokemon are weak to Psychic-type moves. Just use Psybeam, don't waste your Psychic PP.
Lance: Make sure you are a higher level than Aerodactyl, as its Hyper Beam will KO you and it's just faster than Alakazam at the same level. With that being said, you should sweep no problem, none of his Dragons are bulky enough to survive STAB Psychic.
Rival: If you've chosen Squirtle, your rival will have a Venusaur, and in that case there is nothing that can stop Alakazam except your rival's Alakazam. If you are a higher level, you should take it down though. Watch out for Exeggutor (Charmander and Bulbasaur only though) as it resist Psychic and will hit you with physical moves or hypnosis. In Yellow, your rival will likely have a Jolteon (if you beat him twice) which is often faster without a large level disparity and carries Pin Missile, which can crit over multiple hits and is supereffective. Still, Alakazam is not truly threatened and should OHKO in return.
Additional Notes: Even though Abra needs to be trained like Magicarp, it levels up to Kadabra at level 16 and it levels up quickly. You can then evolve it immediately to Alakazam if you have a friend and/or a trade cartridge. While not as potent as Alakazam, not being able to evolve doesn't make Kadabra useless, but will need to be 5-8 levels higher to do what Alakazam can reasonably do. As Kadabra/Alakazam, it is self-sufficient after that moment and requires no TM support. It is advisable to run Psybeam/Psychic/Recover with any other move of your choice as this allows Kadabra/Alakazam to have some staying power as well as not being limited by PP.