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A note about my comments: I used a team built around my starter Totodile. I do tend to deviate into other areas though, and I do a lot of typecasting, probably because I played the original games so obssessively. However, especially with so many options by this recent game, I must emphasize that you don't always have to rely on type advantage to win gym battles. Also, it is my personal policy when playing these games for the first time to not rely on outside pokemon obtained in trades (except for Event pokemon and they weren't used in gym battles), so what you see is my comments about my own home-grown team up to the point in question.

Don't use your Chikorita!!!

That's my first line of advice, but duh.

If it is fast you might manage Sleep Powder, but I did that with Butterfree.

My starter was actually Totodile, and she does a fair amount of damage.

Unlike in the first games, you can now get to a patch of grass below Violet City and there you can find Mareep. Level it up to at least level 10 first because then it learns Thundershock. That is your best bet! (So I used a combo of butterfree, totodile, and mareep.)

Or, if you don't wanna use electric, you can either use a Geodude or snap up a Bellsprout and trade it for an Onix from the kid in the first house you get to in Violet City.

Finally, a Haunter is good, at least for putting 'em to sleep and confusing them, and they can't be hit by Tackle. Ghost moves don't work on normal-type birds however.

I used Cyndaquil (lvl 14) and Hoothoot (lvl 10). Hypnosis + Ember combo. I did that in first time, 'cuz it's was piece of pie. whee