It can be complicated. For many trainers, Falkner in Violet City seems to be their first stop. He specializes in Flying-types, and winning gets you a badge that lets you use the Fly HM, though that's all the way in Cianwood.

I don't know grass types very well, but I do know they have a weakness to fire, most notably. But fire types are weak to ground attacks, so you've go that covered. An easy way to think about it is "what are my Pokemon weak to? Do I have something I can switch to that's strong against that type?" It's unlikely that you can have perfect type coverage in a team, but that way you can cover most of your weaknesses.