first you need to tell me which Pokemon game you are playing
The third badge is in Goldenrod City.
the third badge in sapphire is at mauville city(north of slateport city)
Head to lavaridge town for your next badge.
In Vermilion city.
The sixth badgeno actually it the third badge
if you are looking for the third gym badge i suggest Veilstone city.
vermillon city
Veilstone City
Its hard to say -.-
You get your 3rd badge in Veilstone city against Maylene.
it is Mauville gym
mauville city