^
In your case and the OP's, I'd personally recommend going with DS9. Let me remark that DS9 is easily my favorite incarnation, so I'm all biased towards it

. It is followed by TOS, with the other shows far behind.
With that out of the way, there's a reason I'd recommend DS9 to somebody who loves TOS. I honestly feel it's next gen Trek series that is most closely related to TOS. It may not seem that way at first glance, but what I'm getting at is that I've always felt it was produced very much in the same 'spirit' as TOS.
What I mean is that it had a similar way of dealing with its characters, letting them play off each other, care about each other but also fiercely disagree in some situations. It also had a similar approach IMHO in that it would take 'good' people and confront them with really tough choices that would not always allow an easy answer, that would test what it meant to be 'good'.
For what it's worth, I'd also add that the writers and producers on DS9 were self-proclaimed TOS fans. And I think this shines through in the "Trials- and Tribble-Ations" episode which really is a tribute to Star Trek and TOS in particular. I also think it shows how well the DS9 characters fit in with the world of TOS, as I was trying to indicate further up.
I suppose one question would be: What is the best starting point? I'd always recommend watching from the start, but DS9's first season is arguably its weakest even though it has some standouts, "Emissary" and "Duet" in particular.
It might be worth watching "Emissary" to get an idea where the characters are coming from and then move on to standout episodes such as "The Visitor", "Call to Arms" or "In the Pale Moonlight".
Just my two cents.