Backstory: Twenty years before TNG began, the Klingons and the UFP were negotiating a peace treaty. The Romulans and the Klingons were also allied at one point earlier.
The Bandi are interested in joining with the UFP, but may in fact turn to the Ferengi, who Picard claims have found other past associates "tasty."
Then close encounter with the Ferengi and first actual close contact in "The Last Outpost." (Hostile.) Some Klingons try to blame an incident in the Romulan Neutral Zone on the Ferengi. The Romulans are, at the end of the season, "back." The Treaty of Algeron restricts vessels from entering the Neutral Zone.
Data has personally met individuals belonging to 1754 different nonhuman races by season 5 in "Darmok," in which the UFP begins to open communications with the enigmatic Tamarians.
As of the start of season 5, the Romulans have been trying to undermine the Klingon/Federation alliance for "over twenty years."
Vulcan was one of the founding worlds of the Federation. Vulcan legal authorities work on their own; extradition treaties cross outside Federation boundaries.