There were 65 answers before this one, and all of them say sludge bomb is the only thing worth getting on Route 43? This is slightly disappointing.
This team was made to defeat Lance. I didn't worry about Red, because going from Lance to Red is mostly just boring level grinding regardless of your team. I guess you can use Snorlax instead of Tauros and use post-game moves like bounce and stone edge if you want your grind to be slightly less boring. In addition to the restrictions in the original post, I put these on myself because I think they make the game more fun. You will obviously want a different team if you use different restrictions.
The team must have at least 4 battling Pokemon.
Those 4 Pokemon must be from different evolution families.
The battle style must be set.
You cannot select the bag in trainer battles.
Cyndaquil -> Quilava -> Typhlosion @ shell bell/charcoal/choice specs
EVs: 100 SpA
- tackle -> quick attack -> headbutt -> swift -> hyper beam
- ember -> flame wheel -> lava plume -> flamethrower
- fire blast -> focus blast
The starter's job is getting you from New Bark Town to Route 43. More specifically, it must defeat Morty, because that is required to surf on Route 43. Chikorita is obviously the worst for the job, as Bayleef's highest attack stat is only 63, and all of its STAB moves get resisted by Li's, Falkner's, Bugsy's, the rival's, and Morty's Pokemon. Feraligatr works great in the late game, because it learns some moves that are super effective against many Pokemon in the last 3 gyms and Pokemon League. However, this team makes the starter pretty useless after reaching Route 43, regardless of how many coverage moves it learns. Before Route 43, Croconaw's STAB moves are significantly weaker than Quilava's. This is why I think Cyndaquil, which learns fire blast from the Department Store, is the best at getting you from New Bark Town to Route 43.
The fire attacks besides fire blast are Cyndaquil's strongest moves before fire blast, and they remain useful when you want an attack that has high accuracy or PP. Fire blast is for Whitney, the rival, and Morty. Dig covers fire and rock and saves a bit of money on escape ropes. Focus blast covers rock and some other things that resist fire. The normal attacks are there because Typhlosion's coverage moves kind of suck, and there are a bunch of Pokemon that resist fire, ground, and fighting.
This team does not need an electric Pokemon for the rival (despite Quilava's type disadvantage) or a fighting Pokemon for Whitney (despite the fact that Whitney is a meme). If you use only Cyndaquil in every battle before Route 43, then it will earn all the experience, and it can easily go above level 27 or 28 before Whitney. At those levels, its charcoal-boosted fire blast will 2HKO Whitney's Miltank and the rival's Croconaw. Cyndaquil's most difficult battle is probably the rival in Azalea Town, because it's before fire blast. Luckily, the dumb programming makes his Croconaw spam scary face, even after it's already faster than your Quilava, so you have plenty of turns to win with your not very effective flame wheel.
Magikarp -> Gyarados @ silk scarf/mystic water
EVs: 100 Atk
- surf -> waterfall
- flail/ice fang
- dragon dance
This is the Pokemon that you catch on Route 43, not the Lake of Rage. When you surf on this route, it is not possible to find anything between levels 26 and 49, but it is possible to find a level 50 Magikarp. So you can put any Pokemon between levels 26 and 50 in your first slot and use some repels, and it should be pretty easy to find this Magikarp in 3 or 4 minutes. Then use one rare candy, and you have a level 51 Gyarados before the fifth gym. It is very possible that the specific combination of Typhlosion, Gyarados, Heracross, Tauros, Krabby, and Farfetch'd is not the best team in HGSS, but I don't think anyone can seriously argue that any other Pokemon in HGSS is better than this level 51 Gyarados.
Strength is Gyarados's strongest move before waterfall, and it remains useful for covering stuff that resists waterfall. Surf covers stuff that resists strength. Waterfall is Gyarados's strongest STAB move. Dragon dance raises Gyarados's game-breaking physical attack to even more game-breaking amounts. You can use ice fang if you think strength is not doing enough damage against grass and dragon.
Gyarados can solo any trainer battle between Route 43 and Lance using only strength, surf, waterfall, and dragon dance. It doesn't need ice fang, items, or EVs. It doesn't even need to reach level 52. I included the extra unnecessary stuff just to optimize this moveset a little bit. I'd say the most difficult battles (ie. the ones that require the most luck to solo) are Will and Karen. Both of them lead with confuse ray users, so it's dangerous to use more than one dragon dance against them. Both have another Pokemon that can survive any move after one dragon dance and then use hypnosis or stun spore.
Because Typhlosion and Gyarados can handle all of the boss battles, the next 2 Pokemon have 2 jobs: counting toward the 6 Pokemon requirement in the original post, and effortlessly defeating regular trainers. Consequently, I want Pokemon that are not too difficult to catch, can be caught at a high level, do not share types with my other battlers, have a strong attack stat without needing to evolve (or evolves very soon after getting caught), and can learn strong moves.
Heracross @ black belt
EVs: 100 Spe
- brick break
- shadow claw
- swords dance/aerial ace
- counter -> close combat
According to the above criteria, Heracross is one of the best. It can be found at levels 25 to 30 in the trees on Route 47, and its physical attack is 125.
Brick break is an useful STAB move that has more PP than close combat and does not lower any stats. Shadow claw covers psychic and ghost. Swords dance raises physical attack, but I recommend using it only if you're hopelessly addicted to Voltorb Flip for some weird reason. Regular people should use aerial ace, which covers bug. Close combat is obviously a very powerful STAB move.
Tauros @ soothe bell/silk scarf
EVs: 100 Atk
- headbutt -> return
- strength -> earthquake
- zen headbutt
- iron tail/payback
Tauros is another Pokemon that fits the criteria that I mentioned earlier. It can be found at level 20 or 24 in the tall grass on Route 48, and it has 100 base physical attack.
Strength and return are Tauros's strongest STAB moves. Headbutt can help you save PP while it is still stronger than return. Earthquake covers rock and steel, and the other moves cover other things. Note that the TM23 iron tail (and none of the other TMs in this answer) is obtainable only once in the whole game, so I added payback as an alternative move in case you don't want to use the TM23 on this Tauros.
- rock smash
Krabby is obtainable before using any HMs and can learn 5 different HMs without evolving, so Krabby is the single best HM slave in HGSS. One move that Krabby can't learn is waterfall, and that is not a problem because Gyarados uses it.
Of all the Pokemon that are obtainable before Cianwood City, Farfetch'd is the only one that can learn fly and another HM without evolving or sketching. This is why I think Farfetch'd is the best fly slave in this game.
These are the Pokemon and items that I recommend picking up when using this team. They are listed in order by how early they become accessible, but you probably don't want to get them in this exact order. For example, you can buy carbos early, but they're pretty useless until you get Heracross much later in the game.
New Bark Town: Cyndaquil, tell Mom to save money until she buys a silk scarf
Route 32: shell bell, old rod
Cherrygrove City: Krabby
Azalea Town: charcoal
Ilex Forest: headbutt tutor
Goldenrod City: bicycle
Department Store: calcium, carbos, protein, TM15 hyper beam, TM27 return, TM38 fire blast, TM52 focus blast
National Park: soothe bell, TM28 dig
Ecruteak City: dowsing MCHN, rare candy
Cherrygrove City: mystic water
Route 32: heart scale
Route 38: Farfetch'd
Route 42: TM65 shadow claw
Route 43: Magikarp
Lake of Rage: black belt, choice specs
Route 30: exp. share
Route 47: Heracross
Route 48: Tauros
Olivine City: TM23 iron tail
Department Store: amulet coin
Blackthorn City: move reminder
Tohjo Falls: heart scale
Victory Road: TM26 earthquake
Farfetch'd, Heracross, and Tauros also have pretty low catch rates, so you might want a sleep user (like Slowpoke with yawn), net balls, and fast balls to make them easier to catch.
That was the team. Please post a comment if you have questions or you think there are any ways to improve this team.
I want to thank Speedrun.com's HGSS manipless glitchless category for telling me about the level 50 Magikarp when that information was not on Bulbapedia.