I’m a Next Gen baby, so of course I recommend watching every episode. But if you want to bounce around and get a feel, here are my recommendations:
“The Outcast” – Riker has feelings for a person from a gender-neutral alien species who feels they are actually female. You will cry.
As mentioned above, “The Inner Light”. Probably my favourite Next Gen episode.
“I, Borg” – LaForge brings home an orphaned Borg and teaches him to be an individual. It’s helpful to have some Borg background, like “Best of Both Worlds” before seeing this one.
For teh lulz, “Q-Pid” – Q sends the Enterprise crew into a live reenactment of the Robin Hood story.
“Measure of a Man” – a local JAG agent must determine whether Data – the android 2nd officer – is a sentient being, or property of Starfleet. You will cry.
“Darmok” – The Enterprise makes First Contact with a race of humanoids who speak entirely in metaphor.
“Masks” – one of the most visually appealing episodes, in my opinion. Masaka is waking…
Also for teh Lulz, “The Naked Now” – This is episode two (or three, if you count Encounter at Farpoint as two episodes). It’s directly stolen from “TOS: The Naked Time” – you’ll remember Sulu fencing half-naked in the corridors.
“The Offspring” – Data builds himself a daughter. You will cry.
I could go on… Half of the fun in Trek is the random episodes in which you find fairly big stars of today in tiny little roles, like Teri Hatcher from Desperate Housewives as random Ensign in a transporter in “Schisms”.