One of my favorite TV shows. I'll give a list of my top 5 episodes.
The Inner Light - A mysterious probe that emanated from a part of the galaxy that was thought to have been extinct for a long time finds its way to Captain Picard. The probe attaches itself to Picard through an energy beam and Picard collapses. He wakes up on the planet Kitaan, one of the planets from the extinct part of the galaxy. A woman greets him and tells him that he has had a terrible fever and that he has been her husband for many years. Picard still remembers everything about his ship and is very confused. After many years, Picard finally lets go of his former life aboard the Enterprise and starts a family with the woman. During all this time Picard has discovered something about the planet.
Skin of Evil - Not really a good episode but one of my favorites because of what transpires in it. Watch it and you will see what I mean.
The Best of Both Worlds - The main villain of the series, the Borg, making their first appearance since Q sent the Enterprise into deep space in the episode Q Who. The Borg prove to be a powerful enemy and capture Picard during a boarding assault. The Borg then head for Earth at top speed with intentions of assimilating the entire human race. The Enterprise gives chase and devises a weapon that may destroy the Borg ship. But destroying the ship would mean destroying the Captain.
Sins of the Father - Worf is informed that his father has been accused of treason back on the Klingon home world. Being the only Mogh family member left, Worf leaves the Enterprise to challenge the ruling. Apparently, Mogh, his father, gave the Romulans access codes to a Klingon outpost seconds before the shields went down and the base was destroyed. Everyone on the planet died except Worf(or so they say). The Klingon High Council leader is surprised that Worf has come all this way to challenge the ruling. There is more to this than meets the eye.
Tapestry - Captain Picard is injured on an away mission and his artificial heart is failing. As he lies on the operating table, he is visited by the enigmatic Q. Q claims to be God and that Jean-Luc has died. Picard refuses to accept this. Q then takes Picard back to when he first got the artificial heart. We now join a much younger Picard(still played by Patrick Stewart, awkwardly enough)who, unlike his grown self, is a womanizing ruffian with a devil-may-care attitude. Fresh from the academy, Picard was eager for his first assignment. His eagerness is what got him the artificial heart in the first place. Now that he's here again, knowing what he knows now, what will he change about his past?
There you have it. 5 good Star Trek episodes for you guys. Hope you get a chance to check some of them out!