I dunno - why not watch them all?
So that's what we've done. I read some re-watch guides to avoid the real clunkers (and Star Trek has a fair share), and ended up watching about two-thirds of the episodes of TOS and then two-thirds of TNG.
I had assumed that it was only nostalgia that made TOS so great in my memory, and my kids wouldn't like it. To my surprise they loved TOS, and were lukewarm to TNG. They said they preferred the strong characters in TOS, and the tensions in the relationship between Kirk, Spock, and Bones. They also found a lot of TNG episodes boring.
For my part, I have a lot more appreciation for TNG than I did when it originally aired - I only watched the first couple seasons at the time. It's still tea-and-an-afghan-blanket TV compared with TOS. I loved the lurid, pulpy tone of TOS, with the eerie lighting and bombastic music. When I think of TNG what comes to mind is ambient ship noise and beige lounges. But when TNG is good it's very, very good. The best 20 or so TNG episodes are great SF and great TV. And Stewart is the best actor to appear in any Star Trek series, by a wide margin.
I've now moved on to DS9 without the kids. I only ever watched a handful of episodes when they initially aired, so I'm going in pretty much cold. Halfway through season 3 now, and I can see why it's held in such high regard. There's definitely more complexity and subtlety than the other series. I thought I'd hate how the Ferengi are featured, but Armin Shimerman makes Quark one of my favourite characters.
However, watching DS9 immediately after TNG has marred my enjoyment of DS9 for a couple reasons:
1) The digitally remastered TNG in HD looks great. Reverting from that to the SD of DS9 sucks. The sets look so dim and soupy.
2) I'm not a fan of Nana Visitor as Kira. She's a decent actress, but I just don't buy her as a hard-bitten, former terrorist. When I found out Michelle Forbes was originally offered the role (presumably resuming her role as Ro) it made Visitor even more disappointing. Forbes would have been perfect.
So far DS9 hasn't approached the best episodes of TOS or TNG in quality, but I'd say there are fewer clunkers too. And I understand things really pick up in seasons 4 and 5.