Yes and no.
Stargate SG-1 follows
Stargate... but from a fan fiction point of view.
Stargate SG-1 is not a direct sequel because of different directors. The original directors of
Stargate might make their own sequels that has no relation to
Stargate SG-1.
It's like this... Director A makes
Stargate. Director B watches
Stargate and says to himself... "Hmm... that was a good movie. I think I'm gonna make a series that follows a few years after the movie."
Rumor has it that the original directors from
Stargate are thinking about making
Stargate 2 and
Stargate 3 as part of
Stargate Trilogy.
Stargate 2 is supposedly to take place 12 years after
Stargate. This will be a different story and in no relation with
Stargate SG-1.
Anyways, the end of the 7th season of
Stargate SG-1 reveals something that spawned a second series...
Stargate Atlantis. Season 8 of
Stargate SG-1 and Season 1 of
Stargate Atlantis follow side by side. If you watched the episodes as they were shown on television, you would notice some episodes cross paths. For instance, the last episode of the 8th season of
Stargate SG-1 leads to the first episode of the 2nd season of
Stargate Atlantis.
After
Stargate SG-1 ended, they decided to make three straight-to-DVD movies that follows
Stargate SG-1 and will likely spawn a third series where the 9th chevron is used. Unfortunately, that new series was put on hold due to the strikes that's been going on.