Abra is available on Johto Routes 34 and 35, and can also be bought from the Goldenrod Game Corner. In Kanto, it is available on Routes 5, 6 (G/S only), 8, 24 and 25 (G/S only). If you encounter Abra in the wild, it will always Teleport away on the first turn of battle.
Abra is an absolutely amazing special sweeper in every single generation, and this is no different. You can catch Abra before the third gym, on Routes 34 or 35, but the much more reliable way to acquire this psycho killer is by dropping 100 coins at the Game Corner. Literally as soon as you obtain Abra, you can walk over the Goldenrod Market and purchase the three elemental punches TMs, which are special in this generation and are therefore fantastic coverage options for Abra. It is amazing how many things just die to the Elemental Punches early-game (as well as mid-game and late-game). Unfortunately, the Abra line has terrible defenses, which makes it a liability against the handful of foes which it cannot outspeed and OHKO, such as Whitney's Miltank, Clair's Kingdra, and Karen's Houndoom. Overall, though, Abra is easily the best pure sweeper in all of GSC, and as long as you can figure out which moves will OHKO and which moves won't, Abra will benefit any team immensely.
Gym #3 - Whitney (Goldenrod City, Normal-type): Kadabra can probably muscle past the Clefairy if you need it to, with Confusion or one of the Punches, but you definitely should not leave it in against the Miltank. It's too strong.
Rival (Burned Tower): Kadabra can sweep through everything. Just use the right Punch move. Maybe avoid Quilava, since Kadabra doesn't have a super effective move to use against it and it can hit back for some serious damage.
Gym #4 - Morty (Ecruteak City, Ghost-type): This fight really depends on whether or not you can evolve Kadabra into Alakazam. If so, then you should be able to spam Confusion/Psybeam without any trouble, but if not, then Gengar will probably take you out with Shadow Ball. Gastly and Haunter should be safe for both, though.
Eusine (Cianwood City, Crystal only): The Drowzee and Haunter should not be problems at all, but watch out for Electrode's Thunder if you still have a Kadabra. Alakazam should tank it well enough.
Gym #5 - Chuck (Cianwood City, Fighting-type): You should be able to reliably 2HKO the Poliwrath while not being KOed by Surf. However, this depends on whether or not you can evolve your Kadabra into Alakazam or not, as Kadabra will probably nail the 2HKO just barely or be a 3HKO instead. In that case, watch out for DynamicPunch.
Gym #6 - Jasmine (Olivine City, Steel-type): Fire Punch will destroy both Magnemites, but you may not want to risk it against the Steelix if you are a little underleveled. Alakazam has no such problem; Fire Punch away.
Rocket Executive battle #2 (Team Rocket HQ): Again, spam Psybeam. Except for that Murkrow... which dies to ThunderPunch or Ice Punch.
Gym #7 - Pryce (Mahogany Town, Ice-type): You should be able to come out on top just fine because Pryce's 'mons are even lower leveled than Jasmine's. ThunderPunch is a winner here, and them not having anything super threatening is a plus. Use Fire Punch for Piloswine.
Rocket Executive battle #3 (Goldenrod Radio Tower): Psybeam spam everything and watch them crumble at your feet... but only fight Weezing if you already have Psychic, just to be sure. Its bulk is not too bad, and an underleveled Kadabra might not score the OHKO.
Rival (Goldenrod Underground): You can take out everything with the appropriate coverage move with the possible exception of the starter 'mon, which is less than ideal if it's Feraligatr, due to being fairly bulky and physically strong. Meganium isn't scary, nor is Quilava for an already evolved Alakazam.
Rocket Executive battle #4 (Goldenrod Radio Tower): Psybeam or Psychic spam are the way to go. For Murkrow, once again, use ThunderPunch or Ice Punch.
Rocket Executive battle #5 (Goldenrod Radio Tower): Koffing is easy as pie, but Houndour and Houndoom are no good. Houndour may be possible for Alakazam without much trouble, but Houndoom is categorically off limits.
Suicune (Bell Tower, Crystal only): Spam ThunderPunch. Suicune is defensive and will take some time to die, but its offensive power is not too high, so Alakazam should counter it well.
Gym #8 - Clair (Blackthorn City, Dragon-type): Alakazam should be able to OHKO all the Dragonairs just fine with Ice Punch, but you should let someone else take care of Kingdra. Its Hyper Beam is no joke.
Rival (Victory Road): Use the appropriate coverage moves and win. Just watch out for the starter, unless it's Meganium; Fire or Ice Punch away.
Elite Four Will (Indigo Plateau, Psychic-type): You get super-effective coverage against every single of his 'mons, so Alakazam should have a really easy time here.
Elite Four Koga (Indigo Plateau, Poison-type): Another cakewalk for Alakazam. Spam Psychic against what's weak to it and use Fire Punch for Forretress.
Elite Four Bruno (Indigo Plateau, Fighting-type): Again, great matchup. Psychic everything that's Fighting and Ice Punch the rest.
Elite Four Karen (Indigo Plateau, Dark-type): Umbreon and Houndoom counter you (gasp!) but you can steamroll through the other three 'mons. Use Ice Punch or ThunderPunch for Murkrow.
Champion Lance (Indigo Plateau, Flying-type): Gyarados and the Dragonites easily fall to the appropriate Punch, as long as you aren't too underleveled. However, Aerodactyl might outspeed you, and Charizard probably won't fall to Psychic or ThunderPunch, so watch out for those two. Unless you're certain to outspeed Aerodactyl, leave it to a physical wall. Charizard should be okay; ThunderPunch hits it hard and it mostly has special moves, which Alakazam is not too bad at tanking.
From this point onwards, you can fight the gyms in any order, though you will need to retrieve the Machine Parts from the Cerulean City gym before you have access to the earlier portion of Kanto. Feel free to anticipate or postpone any battles as needed.
Gym #9 - Brock (Pewter City, Rock-type): Ice Punch everything that isn't a fossil, ThunderPunch that instead. Easy!
Rival (Mt. Moon, optional): Again, the appropriate coverage move will do the trick. Gengar and Alakazam are evolved now, but it shouldn't matter since yours will be higher leveled anyway.
Gym #10 - Misty (Cerulean City, Water-type): Everything except Quagsire is ThunderPunch, and nothing hurts! Take on Quagsire with Psychic if you have a few levels of advantage. Otherwise, don't bother; Alakazam's low Defense would make it risky.
Gym #11 - Lt. Surge (Vermilion City, Electric-type): Alakazam can Psychic whatever is not an Electrode here. The Electrodes are best avoided because they could survive one hit, and then SelfDestruct back, which is very dangerous. However, you might be here with about ten levels of advantage, which makes it matter a lot less. Use Fire Punch for Magneton.
Gym #12 - Erika (Celadon City, Grass-type): Either Ice Punch or Fire Punch everything. You're done. Obviously, use Psychic for the dual Poison-types.
Gym #13 - Janine (Fuchsia City, Poison-type): Psychic, Psychic everywhere!
Gym #14 - Sabrina (Saffron City, Psychic-type): Sabrina's Pokémon are not too tough for an Alakazam above their level. Just pick a Punch move and stick to it. Ice Punch may nail the freeze and make the rest of the battle a breeze, Fire Punch may help you whittle them down with burn damage, and ThunderPunch may impair them with paralysis. Any of them works.
Gym #15 - Blaine (Seafoam Islands, Fire-type): Psychic hits all of his Pokémon neutrally, and it's the best move you can use. Magmar and Magcargo are not too bad, but Rapidash has a strong Attack and a high enough Special Defense to endanger Alakazam if it survives a hit; avoid it if possible. You have a Surfer; use that instead.
Gym #16 - Blue (Viridian City): A somewhat tricky matchup. Alakazam is a mirror match but only has Psychic, so your coverage will get the best of it. ThunderPunch does wonders against Gyarados, as does Fire Punch against Exeggutor. Pidgeot may or may not die to an Ice Punch, and if it doesn't, it could seriously endanger Alakazam; same goes for Rhydon, which is even more dangerous than Pidgeot if it survives. Maybe try Pidgeot, but leave Rhydon to something that can hit it with Grass or Water moves. Arcanine is too strong and has ExtremeSpeed; avoid it.
Rival (Indigo Plateau, optional): Just more of the same. Crobat may or may not outspeed you depending on your level, but it shouldn't be a problem as Alakazam has grown quite a lot by now (unless you're doing this battle before literally every single Kanto gym, which is far from wise). Everything else is just like before.
Red (Mt. Silver): This fight is somewhat tricky as well. You can definitely beat the Pikachu and Venusaur one-on-one, but the other four are more difficult. Blastoise and Charizard are toss-ups, but you should be able to 2HKO them before they 2HKO you, and they are special attackers, which means Alakazam isn't completely uncovered against their hits. Espeon you can beat as long as you can use healing items. Snorlax is a definite no: strong Special Defense, strong Attack, high HP. Nope.
Abra only learns Teleport. As soon as it evolves, at level 16, it will learn Confusion, your main Psychic STAB for a while. Ignore Disable at level 18; you probably already have all three elemental Punches anyway, so it would be a waste of a slot. Take Psybeam over Confusion at level 21. Ignore Recover as well, at level 26, since Alakazam is too frail to use it well. Future Sight at level 31 is a waste of your time and should never be taught to an Alakazam. Psychic finally comes at level 38, completing Alakazam's final set; Reflect at level 45 may be tempting, but it simply does not deserve a slot when there are much better TM options to use.
As for recommendations: just use a STAB attack, Ice Punch, ThunderPunch, and Fire Punch. You can buy all three of them from the Goldenrod Department Store, which you should have already reached by the time you catch an Abra. However tempting Alakazam's various "support moves" may be, they completely pale in comparison to the four attacking moves that Alakazam can use to annihilate as many Pokémon as possible.
Recommended moveset: Psychic, Fire Punch, Ice Punch, ThunderPunch
Minimum stats are calculated with 0 EVs, IVs of 0, and a hindering nature, if applicable.
Maximum stats are calculated with 252 EVs, IVs of 31, and a helpful nature, if applicable.
Abra Line Ratings
At what point in the game should I be evolved? Believe it or not, Abra can still wreck things with the elemental punches before evolving, but obviously it becomes that much more powerful when it evolves. The same applies from Kadabra to Alakazam. If you want to be able to solo Morty, then you probably are going to want to have an Alakazam by that point. It's best to evolve Abra as soon as possible; try to get that done while you're grinding for Whitney.
How good is the Abra line in a Nuzlocke? Abra is really, really good, but not quite great. It can and will streamroll through the vast majority of Kanto and Johto, but you have got to be of the mindset that ANY attack will one-shot Abra. Obviously this is an exaggeration, but really -- if you send Abra in there, you should be 100% sure it can outspeed and OHKO the opponent. If you're ever unsure, then it probably can. The Abra line is certainly not without counters in GSC, but as long as you prepare for those particular threats, then Alakazam is free to be the Psycho Killer it has been since 1996.