During “The next Generation” we are introduced to quite a few alien races, and with them to a lot of new political plotlines.
The neutral, albeit friendly relationship to the Klingon Empire deepens. Even Lieutenant Worf is a part of Starfleet. Orphaned during the Kithomer Massacre he is raised by humans and thus becomes part of this world rather than his own. His character teaches us a lot about Klingons in general and about human/Klingon diplomatic problems specifically. In the beginning of TNG, Klingons tend to look down upon the “weak” Federation members. Later on, mutual respect and a much deeper understanding between each other is achieved.
The relationships towards Vulcan and Romulus stay very much as they were set in the original series.
New races introduced do not really feed a “political megaplot” yet. The Ferengi – very capitalistic traders – while first designed to take over the antagonistic part the Romulans had been playing, remain relatively vague in their motives and only play a small role in general politics. The same goes for almost all freshly introduced species – friend or foe – their general political impact isn’t very high.
For the first time the Federation is shown receptive to corruption. Alien parasites almost manage to take over Starfleet.
The main political plot of “The next generation” of course takes place in the conflict with the Borg. The Borg aren’t a “race” as such. They assimilate and destroy. They are “the” enemy. While assimilated people that have become part of the Borg collective can be returned to being individuals, the process isn’t easy. After Captain Picard’s assimilation and rescue, the conflict turns into open war.
Generally “The next Generation” was a highly political series, even though the political plots generally remain based mostly on single episodes. Freedom of speech, corruption, criticism on capitalistic systems, problems of migration and many more topics play a huge part in the series. I personally can only recommend watching the series with a sensitive eye towards political topics. Many of the episodes are as actual as ever nowadays.