Dragon Ball tells the story of Goku through the end of his childhood and spans 153 episodes. It is the foundational story of Goku and his friends, and skipping it is not recommended. If you want to skip filler episodes, you can use this guide.
Then, watch EITHERDragon Ball ZORDragon Ball Z Kai.
Dragon Ball Z spans 291 episodes. The Japanese voice actors were in their prime when recording Z, and the Funimation home release with subtitles by Steve Simmons is highly recommended. The Funimation dub, on the other hand, is notorious for its inexperienced voice actors and egregiously creative translations. If you want to skip filler episodes, use this guide.
Dragon Ball Z Kai spans 167 episodes in the international version and 159 episodes in the Japanese version. It is a remastered version of Z with the majority of the filler removed. The Funimation dub offers a now-veteran cast and considerably more accurate translations. If you have no nostalgia for the original Z dub, this is the recommended route.
Dragon Ball Super comes next, according to in-story chronology. It begins about 4 years after the final battle of Dragon Ball Z, or about 6 years before the 10-year time skip at the end of Z (episodes 289-291). The first two arcs of Super are retellings of the final two Dragon Ball Z films (14 and 15), so you have a handful of choices on how to continue the story:
The anime-only route: Skip straight to Dragon Ball Super, understanding that it adapts Battle of Gods and Resurrection 'F' into story arcs, with some details changed. For a list of changes, see here and here, scrolling down to "Main Differences From the Theatrical Film".
The combination Super/Movies route: Watch episodes 1 and 2 of Dragon Ball Super. Watch Battle of Gods. Watch Episodes 15-18 of Dragon Ball Super. Watch Resurrection 'F'. See this post, then continue with Episode 28 of Super.
The quickest route (for the lazy/impatient): Watch Battle of Gods and Resurrection 'F'. See this post, then continue with Episode 28 of Super.
Dragon Ball GT spans 64 episodes and begins 5 years after the 10-year time skip at the end of Dragon Ball Z, i.e. well after the time setting of Super. It follows those events, but it is the only series not based on a story by Akira Toriyama, and its continuity has been overwritten by Dragon Ball Super.
The rest of the movies/specials are separate from the main storyline, and are entirely optional. If you'd like to watch all of the movies and specials in the order released, use this list.