Can someone recommend a few sample episodes as a taster? Nothing that will spoil a big story arc if I decide I like it, but a few things that make the show look good, and will persuade my missus it's worth giving this a shot.
Hmm... Interesting. Good sample episodes without huge plot or background knowledge required. I'll pick 3 episodes from TNG, DS9, and Voy which I think show off the potential.
The Measure of a Man - A good court room drama but with a sci fi twist
Clues - A proper sci fi mystery episode
The Drumhead - Another court room episode so perhaps substitute 'The Wounded' for a bit of an action episode
Duet - Might be a bit difficult without any back story knowledge but a terrific episode (alternatively Civil Defense is good fun).
Whispers is in my recent memory and might be a good choice, strong plot and easy to follow
Our Man Bashir - Terrific spy episode showing the breath of what Star Trek can offer
Displaced (on second thoughts, perhaps Message in the Bottle for some humour). I've always had a soft spot for Displaced but it's not amazing on a technical level.
Counterpoint (or Blink of an Eye)
With Voyager there are quite a few easy choices to go for. Voyager is made so you can enjoy the episodes in pretty much any order, there is no strong running strong ark happening and most episodes are quite acceptable/good but few are great/fantastic.
Edit: The above suggestions are pretty good, with my choices I tried to go away from the iconic episodes and towards smaller more self contained episodes. Episodes like Yesterday's Episodes are great, but I think you get the most out of them if you are fairly familiar with the crew already. Ditto with The Best of Both Worlds, you'll enjoy it whatever happens but some of the plot will have more weight if you are more familiar with the characters. The Visitor is a good choice though.