Ok, I will take a shot at this
Johto
- Gym #1(Flying) weak to: Ice, Electric, Rock
- Gym #2(Bug) weak to: Rock, Flying, Fire
- Gym #3(Normal) weak to: Fighting
- Gym #4(Ghost) weak to: Ghost, Dark
- Gym #5(Fighting) weak to: Flying, Psychic
- Gym #6(Steel) weak to: Fire, Ground, Fighting
- Gym #7(Ice) weak to: Fire, Fighting, Steel
- Gym #8(Mainly Dragon) weak to: Dragon, Ice, Electric
Kanto
- Gym #1(Electric) weak to: Ground
- Gym #2(Psychic) weak to: Ghost, Bug, Dark
- Gym #3(Grass) weak to: Ice, Fire, Bug, Flying
- Gym #4(Water) weak to: Grass, Electric
- Gym #5(Poison) weak to: Psychic. Ground
- Gym #6(Rock) weak to: Water, Ice, Grass, Ground, Fighting
- Gym #7(Fire) weak to: Water, Rock, Ground
- Gym #8(Miscellaneous): varying weaknesses
Elite 4
- Will(Psychic) weak to: Ghost, Bug, Dark
- Koga(Poison) weak to: Psychic. Ground
- Bruno(Fighting) weak to: Flying, Psychic
- Karen(Dark) weak to: Fighting, Bug
Champion Lance (Mainly Dragon) weak to:Dragon, Ice, Electric, Water
Water
Although none of the Johto gyms are necessarily weak to the Water type, the water types can take on most of them as they hot everything but Dragon with Neutrality with some exceptions (Chucks's Poliwrath and Pryce's Dewgong and Seel). This counts for Grass and Fire as well, however grass will be facing a tougher time with the first 2 gyms and the 6th.
However having these types doesn't mean that you will necessarily defeat the gyms but it really depends on the Pokemon and the moves they learn. Most Water types learn many Ice type move and usually have access to at least one fighting type move and some even have access to dragon type moves so I would say that you will be getting very far even to the extent of defeating the entire Johto league with a Water type, however this can also be done when you overtrain your Pokemon.
Fire
types are also known for their powerful moves and usually appealing movepool as they have access to very powerful moves including Earthquake like most Pokemon do.
They are usually fast and powerful and very easy to obtain and since almost every fire type in HG/SS is very powerful, you will most likely be wanting to have at least one in your party
The grass type will also be seeing some use but not as much as the first 2 types as Fire types are also capable of learning some Grass type moves themselves. So I would say that a
Psychic/Ghost
or Ghost type will most likely be finding much use in defeating the leagues. They have one of the widest movepools and are usually fast and powerful. Their movepool usually consist of Grass, Electric, Psychic, Ghost, Fighting etc... They can also take on the first 3 members of the Elite 4 like professionals so I would say that they are probably the best types that can take on HG/SS. Team Rocket also tends to have a lot of poison types so they will even make the game too easy to beat
Flying
types are pretty much mandatory as they will enable you transport and can hit a lot of Pokemon for neutral damage as well as cut the life span of fighting, grass and bug type really short. They usually have a relatively nice movepool that consist of mainly flying type moves, however there are usually powerful enough to take out many Pokemon. There are some variants such as Togekiss that learn extremely powerful moves such as shadow ball, extrasensory, shock wave, Aura Sphere, Flamethrower etc... which will be a very interesting asset to many parties
Dragon
Known to be the most powerful types a dragon type is of course the best type that you could have as they have an extremely wide movepool as well as access to devastating and powerful moves. Though they are rare and hard to train, it should be noted that a Dragon type can pretty much take on the entire game with relative ease only having a harder time when facing Lance
Ice
In HG/SS Ice types have a rather impressive presence of utility as they will help you take down the toughest of the Elite 4 with much ease. Despite only resisting itself, they have a nice offensive presence and will make quick work of threatening opponents
Electric
Electric types are extremely useful as they make quick work of powerful types such as Water and Flying. They hit most of the in-game gyms for at least neutral damage and can kill one of the toughest Pokemon to face in-game...Gyarados as well as Charizard. However it isn't necessary to have one slot going to an electric type since many Pokemon have access to a few electric type moves
Ground
Bulky and powerful and graced with the feared Earthquake. Ground types usually have a high attack as well as a very nice bulk. These 2 traits make them feared opponents and outclassers of the Rock type. Wielding a very nice movepool, a ground type will be a great aid when taking on the Elite 4
Bug
Although you are very limited to a few extremely useful bug types and for the fact that they are very rare Pokemon, Bug types are very deadly in HG/SS as they are capable of demolishing nearly gym with a small amount of difficulty, however your choice between them is very small as not all of them are capable of such damage. The one that will most likely be killing this much would be Heracross.
Rock
Rock types have a nice offensive presence however they carry many weaknesses with them. They can be very useful during the first part of the game but the further you reach, the less useful they tend to become. However it should be noted that they have a very nice movepool as well as access to many powerful moves
These are the types that will make the game easier to defeat but overtraining your Pokemon will most likely enable you to defeat the entire game with equal ease
