Stargate SG-1 is the first series. If you plan on getting in to Stargate, you should start by watching the first season of this, as it establishes the rules of the universe. After the final season, they released 2 direct-to-DVD movies, Ark of Truth and Continuum, which should be watched in that order. This show is overall a light-hearted sci-fi adventure across the galaxy. Sometimes things take a serious tone, as it's not all fun and games.
Stargate Atlantis is the second series. It started airing at the same time as SG-1's 8th season. Each week, they aired an SG-1 episode alongside an Atlantis episode. After the pilot episode, there's a fair amount of crossover between the two shows, as the two shows are taking place in the same chronology in universe during this time. But it's not necessary to watch both series at exactly a 1:1 episode rate, but it helps to be aware of the goings-on in the other series. This show maintains the same tone as SG-1, but with different dynamics within the central team.
Stargate Universe is the third and final series. It's set after SG-1 ends, and its premise depends on a status quo in the Stargate universe that was established over SG-1's run. It started airing after Atlantis ended (SG-1 ended a few years prior to that). There's almost no crossover with this show and the prior ones, as part of the premise of the show is to explore untouched areas in the Stargate universe with a darker tone to the storytelling. Characters from the other shows appear maybe 3-4 times total.
That's a solid overview of the franchise, let me know what other details you're seeking, and I'll add them.