Well, to beat Brock easily with a Charmander team you can either level Charmander up to level 13 (?) and get Metal Claw which is super effective to Rock types or you can go back to Viridian and go west to the Pokemon League and in the grass near the entrance you can get a Mankey which is Fighting type so it can beat Rock (later in game I would suggest replacing for a Machamp). Those are good for the start of the game (Brock) but once you get to Misty then you get more choices.
Don't even bother trying to use Charmander. Get a Bellsprout which will easily sweep Water types if you level it up (Personally I think that Bellsprout's evo line is good so this is a good Grass type to have) or you could just get a Pikachu, always a good one but I prefer Jolteon which you get later in the game :).
Obviously for Lt. Surge catch a Diglett from Diglett's cave and you'll have no problem (I wouldn't keep Dugtrio but if you want a Ground type always a good stand by).
In Lavender town I suggest getting a Ghastly but I'm not sure if you can catch them yet :/ they can be quite useful later on.
Next is Erika which you can easily take the entire gym with your Charmeleon or Charizard.
Now Saffron City, for the Fighting Dojo you can use pretty much anything but Normal types it's not that hard. Now the real gym - If you didn't catch a Gastly in Lavender town I suggest you go back and get one. They are super effective to Psychic types and Gengar (Ghastly's final evolution) is beast.
For the next gym I suggest you go and get an Abra (near Cerulean) and evolve it. Kadabra and Alakazam are great Pokemon and will sweep this gym.
The next gym (Cinnabar) is fire so Lapras, Gyrados, Starmie are all good choices I would say Starmie or Lapras because Gyrados's stats are Attack heavy and it has a special movepool. Starmie has Psychic as well but Lapras is better over all but not for this gym.
The next is Giovanni he likes Ground types so Lapras is VERY good here - just with Ice Beam and Surf it could destroy this gym.
So for Gym's
Brock - Leveled up Charmander with Metal Claw/Mankey (Pokemon League entrance)
Misty - Pikachu (Viridian Forest)/Bellsprout (Route 24)
Lt. Surge - Diglett (Diglett's cave)
Erika - Charmeleon (leved up Charmander)
Sabrina - Gastly/Haunter (Pokemon Tower - Lavender Town)
Koga - Kadabra/Alakazam (Evole Abra from Route 24)
Blaine - Lapras/Starmie Probably Lapras (given to you in Silph Co)
Giovanni - Same as above ^
I hope this is enough to get you started I will do an Elite 4 version next.
Elite 4 time
The First is Ice and Water heavy so having a Charizard is a kinda bad thing to have because Surf or Ice Beam can know one out. Grass Types can take the water or Electric, I would suggest Electric because it isn't weak to Ice or Water but can take out both reasonably well.
Next is Rock/Fighting. Flying types can take down the Fighting and Water can take down the Rock. Also Psychic is good.
Now it's Ghosts. A Gengar might be able to do this with good speed and attack because Ghost is super effective against Ghost. If Gengar dies Charizard might be the one to do the job with good offence :)
Now is Lance he likes Dragon types. Ice can finish all of these of. So a Lapras with Ice Beam should do it.
Now for Blue he has different types so he is stronger than the rest. So I can't say a specific type for him though a really high level Lapras could take it no problem.
Here's a team I would suggest for just doing it, no favoritism.
- Charizard (because that's you're starter and can be useful to just attack or stand by to sack (let it be attacked so you can revive/heal a better Pokemon))
- Lapras (Lapras is amazing against Blue super effective to 4 of his 6 Pokemon and is good against the Elite 4 especially Lance and Bruno)
- Gengar (Strong against Agatha will do ok with the other E4)
- Scyther (IMO, an underated Pokemon. Set up a few swords dances and get it Aerial Ace and it can do loads of Damage and when you have the National Dex EVOLVE IT TO SCIZOR , will do well against Fighting types and A few other Pokemon)
- Jolteon/Raichu (pretty average, will do well against anything that it's not weak to, good for Lorelei)
and the final Pokemon Pokemon might be either of these two.
- Victreebell (Mixed attacker, use to put Pokemon to sleep and others finish them off or as a grass type)
- Alakazam (Great Special attack use on anything but Ghosts)
I would say Alakazam.
:) have fun.
Remember don't just do what I say take other peoples opinions into account and which ever Pokemon YOU LIKE, YOU USE as long as they aren't things like Caturpie or Magikarp..