I am definitely a big star trek fan so I will add my thoughts
Series:
1) Original Series- I grew up watching this series as a kid. Nowadays, most of the episodes seem... silly. But for nostalgia sake I still really enjoy them, the characters are overall still my favorites, and some of the classic episodes are definitely still my favorites. Even though overall I think the next two are better, I always put this one first just because.
2) Next Generation- I also saw some of this as a kid, but not as much. When I was really little I thought my grandpa was captain picard (because they were both bald) but I figured out when I was older that this wasn't the case (I think my mom broke the news to me). I also loved the series itself, it was the flowering of the Original Series on a good budget and with "better" actors.
3) Deep Space Nine- some days this one winds up in the number 1 or 2 spot, because as a collective series I enjoyed the story so much. I only watched this one within the last 5 years. I loved how much the characters and the setting changed throughout the story, it really
went somewhere, and I hope the new Star Trek Discovery series does something similar. Loved the characters too.
4) Voyager- Watched this one a few years ago, right after Deep Space Nine, but overall was disappointed. Some of the episodes and characters were great; it was really trying to go back to the roots established in the Original Series but with a more modern approach, but parts of it were stale to me. Didn't care for it much, despite the Janeway character being good.
5) Enterprise- didn't care for it, other than the "pre-original-series" nostalgia easter eggs, and the andorians.
Episodes:
1) Inner Light (TNG)- the one where Picard lives out his life on an alien planet and grows old.
2) Space Seed (TOS)- first appearance of KHAAAAN, most awesome bad guy of the original series, so good they made a movie for him too.
3) Mirror Mirror (TOS)- alternate reality where Spock has a beard. Also Uhura. Need I say more.
4) All Good Things (TNG)- two parter, final episodes of the Next Generation. Giant Q mind-screw across multiple time periods.
5) Trials and Tribulations (DS9)- The Deep Space 9 crew go back in time... and see Kirk and the Enterprise! And tribbles, huge nostalgia episode that I loved.
6) Living Witness (VOY)- my really only favorite Voyager episode. 700 years in the future after the Voyager visited their planet, the aliens come to realize that history is not the same as they remembered it, thanks to a surviving backup of the Doctor's hologram.
7) In the Pale Moonlight (DS9)- amazing episode, a conspiracy is underfoot. Can't say more, its awesome and a great story.
8) Chain of Command (TNG)- THERE ARE FOUR LIGHTS!
9) Darmok (TNG)- Shaka, when the Walls Fell. Interesting take on extraterrestrial communications. Fascinating episode.
10) Badda-Bing Badda-Bang (DS9)- the crew of DS9 helps out their holodeck friend Vic Fontaine (basically an Oceans 11 kind of episode, but it was quite enjoyable).
11) Arena (TOS)- its so campy and bad, but it has the gorn fight with Captain Kirk. Love it still.
Movies:
Overall I didn't care for any of the movies as much as I enjoyed watching the tv series themselves, but these are my favorites.
1) First Contact (TNG)- has the borg, Picard is brought to his breaking point, and time travel of a momentus event. Also technically the start of the Enterprise series, lost of footage was used from this episode.
2) Wrath of Khan (TOS)- because Kirk and Khan.
They fight.
3) Undiscovered Country (TOS)- Kirk and friends defeat a Russian conspiracy and help the thaw of the cold war.
4) Star Trek (2009) (TOS Kelvin Timeline)- I really enjoyed this movie, definitely more than the most recent two, but not as much as the originals. Very pretty to look at, the opening with the Kelvin was AWESOME.
5) Insurrection (TNG)- it was ok, but didn't start that great. Better than the rest in my opinion though.
Characters:
In no particular order:
Kirk, Spock, McCoy, Scotty, Uhura, Sulu, Khan
Picard, Data, Q, Guinan
Quark, Sisko, Garek, that evil Cardassian guy who kept switching sides the entire series (edit: Gul Dukat)
That Andorian guy from Enterprise (edit: Shran)