First of all, as you probably know, those who are talking about dark-types and Wobbuffetts in Red are retarded. Even more retarded is that guy telling you to quit playing Red and switch to the newer versions: obviously, if you wanted to do that, you already would have. That's like going on the "baseball" section of answers.com just to tell people to forget about baseball and watch/play basketball instead.
Mew: You really should give it Softboiled. In case you didn't know, this is like Recover but with half the PP, but you can use it out of battle to give hp to your other pokemon. It's really an awesome move, and the only possible reasons for Mew to be without it are Swords Dance+Explosion and Dream Eater. Also, you might consider giving it something other than special moves: Mew's not bad at that, but Mewtwo does it better. Mew, being able to learn every TM and HM plus Transform, has much more versatility (which you are using a little bit: Mewtwo can't learn Surf) that it should use to do things no other pokemon can and take opponents by surprise in battle.
Arcanine: Not bad, but be advised that Take Down sucks because Double-Edge is better in every way.
Dugtrio: Earthquake and Dig are redundant. You should probably only have one. This is a common mistake that costs pokemon players big time. There's not much point in one pokemon having two moves of the same type, especially when it can only learn four moves total. By giving a pokemon two moves of the same type, you sacrifice 25% of its versatility and gain almost nothing. Imagine you get to pick four tools to do an unknown job: Would you really pick two hammers, just because you're good at using them? Earthquake is offensively better since it hits right away, but Dig has its uses, such as making opponents take exra poison damage and dodging Explosions. Since you already have Dig on Arcanine, I would recommend having Dugtrio keep Earthquake. An obvious move to take Dig's place is Rock Slide because flying types (which are immune to ground) and bug types (which are resistant to ground) are weak to it, but there are other options too.
Jolteon: Same deal, there's not much point in having both Thunderbolt and Thunder. Also, Thunder Wave is only good in Pokemon Stadium. On gameboy, both in one-player and linkbattle, I always hope my opponent will use Thunder Wave. Why? Because Paralysis is not nearly as devastating as Sleep or Poison (In Stadium, the opposite is true). When something of mine gets paralyzed, I say hooray! It just got immunity to sleep, poison, burn, and freeze. Good options for replacement moves include Double Kick, Reflect, Toxic, Mimic, Substitute, Double-Edge, Hyper Beam, Double Team, among others.
Gyarados: Dragon Rage is an awesome power at lower levels, but when your pokemon grow up it seems like child's play. In the 40s-50's it is still a somewhat viable move (though weak compared to Seismic Toss and Night Shade), but keep in mind that as your levels advance (provided those of your opponents do too), you will want more and more to replace that weak Dragon Rage. You might want to replace it with an Electric or Fire move, or perhaps with something like Reflect, Mimic, Substitute, or Toxic, or perhaps with something else.
Pidgeot: Idk wtf ur talking about with this. Pidgeot can't learn Mega Drain, unless you cheat, and I mean REALLY cheat, unlike what I do with my gameshark. Anyway, there isn't much reason to have both Fly and Sky Attack. In the L50's, I think L54 or L56, it will learn Mirror Move, so you can delete Sky Attack for that. And replace that illegal Mega Drain with a normal move, like Swift or Double-Edge or Hyper Beam (the Hyper Beam of most normal types, including Pidgeot, is even stronger than the Hyper Beam of Dragonite!)
Levels: Well, the higher, the better. So you should get them all to L100.
Weaknesses: Your biggest weakness is Ice. You are also pretty weak to Rock and Water. If I had to annihilate your team, I would feel quite confident and a little bored.
Resistances: You've done a pretty good job of making sure that, in general, nothing will have an easy time defending against your team. You can hit the weakness of any pokemon except a normal type (perhaps teach Submission to Mew?).
Other: One thing your team is missing is status moves. In general, to be as effective as possible, Gameboy teams should have three users of Toxic OR three users of a sleep move (Stadium teams should have three paralyzers). Status moves are great, not just because they hurt and hinder your opponent, but because they set the stage for a lot of synergy to occur with many other types of moves. Poison has synergy with Wrap, Bind, Clamp, Fire Spin, Dig, Fly, Bide, Leech Seed, moves that can cause flinching, moves that restore your hp, moves that increase your defense or special, moves that confuse the opponent, moves that lower the opponent's accuracy, and moves that increase your evasion, and others. Sleep is totally devastating to affe