Mankey can be found with a 10% encounter rate on Route 3, a 5% encounter rate on Route 4, a 45% encounter rate on Route 22, a 30% encounter rate on Route 23, and a 15% encounter rate on both floors of Rock Tunnel. Primeape can be found with a 5% encounter rate on Route 23, a 10% encounter rate on the second floor of Victory Road, and an 11% encounter rate on the first floor of Cerulean Cave.
Overall, Mankey and Primeape tend to fall into the category of Pokémon that are really useful and strong early on, but later, their power and durability tends to fall behind. This stems from two main facts. Firstly, Mankey and Primeape's best stat is Attack, and all the other stats range from mediocre to outright bad; this is a fairly large problem, considering their Attack isn't even that great to begin with. Secondly, they learn very little apart from Normal and Fighting moves without wasting valuable TMs or move tutors that would be much better off with other Pokémon, and even then a lot of the time their main coverage option is Dig, meaning the Gastly line is completely unaffected by anything they can do. However, they do do fairly well early on, and this sweet spot lasts a lot longer than it does for Pokémon like Beedrill and Raticate. Decent, but nothing special, especially considering that many Mankey will end up dead before they evolve.
Rival (Route 22, optional): Do not send Mankey in against Pidgey. It will fail to do any significant damage and the Pidgey's Gust will do a ton to it. Against the starter, it has a decent shot against any of them, though it has a much easier time with Bulbasaur since it doesn't have any special attacks. Overall, though, you're safer slugging it out with a Pidgey or Rattata.
Gym #1 - Brock (Pewter City, Rock-type): This is one of the few spots in the game where Mankey can really shine. If you took Charmander and don't feel like grinding all the way to get Metal Claw, Mankey is basically as good of an insta-win button as Bulbasaur or Squirtle, since Low Kick has maximum power against Onix and is super-effective, and Geodude is pretty heavy too.
Rival (Cerulean City): Karate Chop will do quite a bit to Rattata, but watch for Hyper Fang- it'll almost definitely deal over half Mankey's health. It can also take out Abra, and there ends the list of Psychic-types Mankey can handle. Pidgeotto is once again a no-go, and the starters are all a lot more difficult to handle since they have actual movesets now. Still, Mankey can probably beat Charmander and Squirtle, but Bulbasaur is not recommended since it resists Fighting.
Gym #2 - Misty (Cerulean City, Water-type): Just no. Starmie may not have any Psychic attacks, but your best attack here will be Scratch or Fury Swipes, and Water Pulse off 100 Special attack is very likely to straight-up OHKO Mankey. Avoid this situation if at all possible.
Rival (S.S. Anne): This is where Mankey is really starting to become a burden. Raticate can probably be taken out with a one-two strike from Karate Chop, but Hyper Fang will do a TON of damage considering Mankey's terrible defenses, Pidgeotto is again way too risky, Kadabra knows attacks now meaning Mankey has no chance, and the starter now has decent bulk and strong attacks- it's very unlikely Mankey would win a fight with them without a lot of luck.
Gym #3 - Lt. Surge (Vermilion City, Electric-type): Oh, Surge, if only you had a Magnemite. Mankey would suddenly be so much more useful in this fight. As it is, it can probably beat Pikachu and Voltorb if you decide to teach it Dig, but honestly, get a Diglett. This battle isn't really Mankey's fault, since it's difficult to be more useful for Lt. Surge than Diglett.
Rival (Pokémon Tower): Mankey should be evolved by this point. Once again, Primeape stands very little chance against Pidgeotto, though with improved stats it can do something against it in a last resort situation. Growlithe and Gyarados both have Intimidate, meaning Primeape's damage output will be hilariously low, although if you don't switch in directly and used your Rock Tomb TM on it, it can probably beat Growlithe. Not Gyarados, though. Kadabra is lolnope, Exeggcute doesn't have any Psychic attacks but is annoying with Leech Seed and Hypnosis, and the starter is a bad idea if it's Ivysaur but okay if it's Charmeleon or Wartortle.
Giovanni (Rocket Hideout, Ground-type): The only real chance for Primeape to excel since Brock. Onix and Rhyhorn will both be about OHKO'd by Low Kick, and Kangaskhan will not enjoy a Fighting move either. However, 95 base attack and STAB Mega Punch will do quite a number on Primeape.
Gym #4 - Erika (Celadon City, Grass-type) Don't. Unless you've used your Aerial Ace TM on it, two of Erika's Pokémon will resist Primeape's main attack and the one that doesn't has huge defense. Giving it Aerial Ace gives it good reliability against Victreebel, who has piss-poor defenses, but Vileplume and Tangela will be able to absorb a couple of them and hit back hard with Giga Drain.
Gym #5 - Koga (Fuchsia City, Poison-type): Hahahaha no. Primeape's only good move against Poison-types is Dig, and three-quarters of Koga's team is immune to it. Muk can also set up Acid Armors and Minimizes on the charge turns. Use something else.
Fighting Dojo (Saffron City, Fighting-type): Aerial Ace is a weak move, and just because a move is super-effective does not mean it’s an automatic OHKO. It’s better to just use a Psychic-type or Flying-type here, or even to just skip it altogether.
Rival (Silph Co.): Again, no. Pidgeot will quickly end Primeape's life, Exeggcute has a Psychic move now, Gyarados Intimidates you and murders you with Tackle and Dragon Rage, Growlithe would probably lose but it still irritating with Intimidate, Alakazam... could actually be beaten if you can kill it in less than two turns since its only attacking move is Future Sight, and the starters are no longer easily handled since Charizard has Wing Attack, Venusaur resists Fighting, and Blastoise has too much Defense and HP to overcome without overleveling.
Giovanni (Silph Co., Ground-type): Rhyhorn is easysauce, Kangaskhan is threatening but can be taken down, and Nidorino and Nidoqueen are no-gos if you lack Dig. If you do have Dig, they can ostensibly be beaten, but Nidoqueen in particular has enough bulk and attack that she can probably live one and do a ton of damage with Body Slam.
Gym #6 - Sabrina (Saffron City, Psychic-type): If you really need someone to explain why this is a terrible idea, you shouldn't be nuzlocking in the first place.
Gym #7 - Blaine (Cinnabar Island, Fire-type): Fire Blasts from anything but maybe Growlithe and Ponyta will almost definitely OHKO Primeape. If you get lucky and they miss, Dig can probably take down Growlithe (assuming Intimidate is not in play) and Ponyta. Arcanine and Rapidash are probably too hot to handle, though.
Gym #8 - Giovanni (Viridian City, Ground-type): Oh, hey, a Giovanni fight where Primeape can't murder everything! All his Pokémon have Earthquake, so Dig is a no. His Rhyhorns are still easily taken down by Cross Chop or Low Kick, but Dugtrio, Nidoking, and Nidoqueen are best left to someone else.
Rival (Route 22, pre-Elite Four): By now, the basics of the rival fights should be established. Pidgeot, Alakazam, and the starter are all one-way tickets to a dead Primeape, Gyarados wrecks your shit, Growlithe is pathetic, Exeggcute doesn't have a Psychic attack any more but it still isn't a great idea to make Primeape fight it, and Rhyhorn is free EXP.
Elite Four Lorelei (Indigo Plateau, Ice-type): You can... maybe beat Dewgong? That's about it, though. Cloyster has too much of a Defense stat for Fighting attacks to do much of anything, Jynx is too fast and hits too hard for Primeape to handle, Slowbro resists Fighting and it's been established why this is a bad idea, and Lapras is actually probably done in by a Low Kick but if you've gotten rid of it or it doesn't OHKO, Primeape will not enjoy taking Ice Beams from such a high Special Attack stat. Avoid this fight if you don't want to be facing the E4 with five Pokémon.
Elite Four Bruno (Indigo Plateau, Fighting-type): The Onix just die like chumps, but everything else can handle Primeape easily. Hitmonlee outspeeds and deals a ton of damage with Brick Break and Mega Kick, Hitmonchan can take a hit and OHKO with Counter, and Machamp laughs and sets up Bulk Ups in Primeape's face before killing it with Cross Chop.
Elite Four Agatha (Indigo Plateau, Poison-type): Just plain no. She has three members of the Gastly line, all of which are immune to Primeape's three favourite attacks: Strength, Dig, and your Fighting move of choice. She also has a Golbat, which isn't immune to Fighting but doubly resists it. Really, only Arbok can be beaten without wasting the only Rock Slide tutor in the game (which is much better off with something else) and even then Sludge Bomb is painful.
Elite Four Lance (Indigo Plateau, Dragon-type): Lance claims to be a Dragon specialist, but he's really more of a Flying-type user, which should give you an idea of how good Primeape is in this fight (hint: it's not). Even the Dragonairs should be handled with an Ice user, and if you don't have one, you're probably going to lose to Lance anyway.
Champion Rival (Indigo Plateau): Do we even need to go over this again? Rhydon is easy. Nothing else is. Arcanine and Exeggutor joined Gyarados in terms of threat-level in that they now have stats, meaning Primeape can no longer handle them, and Alakazam, Pidgeot, and the starter are all too strong for Primeape to beat. You don't even have any real reason to use Primeape against Rhydon, since you should have at least one Grass or Water move now.
Post-Game: It should be obvious that taking Primeape to Mewtwo is a terrible, terrible idea.
This is kinda what causes Primeape to be so useless late game. Mankey will start with Scratch, Leer, and potentially Low Kick (level 6). As it grows, Karate Chop (level 11) and Fury Swipes (level 16) are both nice additions. However, the same cannot be said for Focus Energy (level 21) and Seismic Toss (level 26). Rage, which Primeape learns upon evolution (level 28), is a joke. Cross Chop (level 35) is the strongest Fighting STAB you're going to get, but the accuracy is shaky at best. Swagger (level 44) and Screech (level 53) are both gimmicky options that may seem good at first but actually suck. Thrash (level 62) is worse than Strength in most situations since it locks you in.
Speaking of Strength, it's probably the best coverage move Primeape learns. Brick Break is available through TM for people who don't like Karate Chop's lack of power or Cross Chop's lack of accuracy. Dig is also learned through TM, and lets Primeape actually damage most Poison-types. Aerial Ace is weak, but potentially worth it for coverage on Bug and Grass types. Earthquake is possible, but honestly, Primeape is going to be falling far behind by that point. Why would you waste such a powerful move on such a mediocre 'mon? Rock Tomb will be mentioned solely for the fact that you weren't going to use it on anything else anyway, so why the hell not? It lets you beat a couple Flying-types that don't have Flying attacks (aka Zubat). If you want a stronger version of that, you could try Rock Slide, but that’s just a waste of a Move Tutor.
Recommended moveset: Cross Chop / Brick Break, Dig, Strength, Aerial Ace / Low Kick / Rock Tomb / Rock Slide
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.
Mankey Line Ratings
What Nature should I be looking for? Adamant is the absolute best possible. Failing that, anything that is +Attack or -Sp. Attack is decent. You probably shouldn’t use anything that is -Attack (Attack is its one big strength, and it needs as much as possible) or +Sp. Attack (the one stat it can afford to lose points in).
At what point in the game should I be evolved? Before Lavender Town at the least, ideally around Lt. Surge.
How good is the Mankey line in a Nuzlocke? Honestly, not that great. They have a severe tendency to be done in early on due to their terrible defenses, which makes them generally undesirable, not to mention the fact that they're fairly rare anyway.