Absolutely do not watch Enterprise first. It’s chronologically earlier, but it’s the last-produced series, and it mostly sucks. (The last season and one or two earlier episodes are quite good, but the bulk of S1-2 are just boring, with S3 only somewhat better.)
TOS is obviously the original. It’s great, but it’s certainly not paced like a modern TV show, and it’s tough to swallow more than one in a single sitting.
The animated series was a kids’ cartoon version of TOS, made about a decade later. It’s fun, but a trifle.
TNG takes place not quite a century later and is probably the best series for someone to get hooked on. It and DS9 can credibly claim to be the best series, although for different reasons. It has the best single episodes, and it’s pretty much an unbroken string of good ones from S3 through S6. So I’d say start there, and it’ll make you a Trekkie. Then you can go back and watch the first two seasons (which were dire and trivial, respectively) and S7, which ain’t bad, but isn’t nearly as good as it was.
DS9 comes next (it started during the middle of TNG’s 6th season), and it’s fantastic. It’s still a little shaky at the beginning, but takes much less time to find its footing than TNG, and it’s much more sophisticated in using characters and building stories over years. But it’s certainly less friendly to the casual viewer, and, with few exceptions, I wouldn’t say it’s episodes taken as themselves are as good as the best single TNG episodes. (Even if the effect can be greater to someone who knows all the backstory.)
Voyager came next, showing up about halfway through DS9’s third season. It is well done sci-fi action/adventure, but it’s not really the fiction of ideas. It’s very supportive of the casual viewer – to the detriment of the regular viewer, in fact, because they never even reference stuff from past episodes that people in these situations would talk about, their use of resources makes no sense given the situation (trapped alone, far from any ability to resupply), and characterization is spotty. But pretty much every episode is an enjoyable yarn. Lots of people whose first Trek show was Voyager love it for its consistent effervescence, but those of us who were around before it rarely do. But for what it is, it’s extremely digestible.
Enterprise came last, the final series, premiering a few months after Voyager’s bow. It was meant to be a prequel, but the people in charge for the first three seasons didn’t actually know that much about Star Trek, so it mostly made the Trek faithful’s heads explode with how often it got shit wrong. And Futurama jokes aside, it really did [almost] kill the franchise. The first few series are lame, and there’s a time-traveling metaplot that makes no sense, clearly wasn’t thought through ahead of time, and in fact, never really gets resolved. (But that’s OK, it blew.) As I mentioned, the third season picked up some steam with a heavy-handed War on Terror parallel, and S4 is excellent. It (finally!) plumbs Trek history deeply, however, so it’s better if you’re a Trek nerd.
So anyway, to rbeak it down – watch TNG’s seasons 3 through 6. You don’t need to know anything beforehand to appreciate it. Then move on to DS9 from the beginning and give it a little patience. It’s certainly the most intellectually ambitious of the group. Voyager and TOS are both fun to watch and don’t require a ton of investment, but neither is the best introduction for a modern audience. Enterprise save for completism’s sake, although prepared to be pleasantly surprised by S4. (The last episode’s awfully rough, though.)