Start with the original series (ToS), and watch in publication order, excepting the animated series. This is also (excluding time travel stories) chronological order, except for Enterprise.
You should include the movies (except perhaps the first, The Motion Picture, and the reboot, Star Trek (2009)). The movies are really just 'feature length' episodes, and you will want to have seen them to understand what has happened in the overall storyline. Some of the movies, particularly the TNG ones, are weak, but so are some episodes, and I'd still recommend watching all of those.
I recently watched all of the series (ToS, TAS, TNG, Deep Space Nine, Voyager, Enterprise, and the movies) in this order. It took a while, but it was worth it, and I understood more of the context when earlier events were referred to. (In my case, I had watched much of TNG, some episodes of ToS and DS9, and a season of Enterprise, previously, but I don't think that had a significant effect on my enjoyment of this order).
The original series is clearly dated in terms of effects (it was produced in the late 60s), but there are some very good episodes, and it sets up the universe, especially the significant events between current day and then. There are only 80 episodes (79 if you skip the original pilot, The Cage, which is substantially different) and 6 films, so it doesn't take long to get through these and into TNG.
It's worth starting The Next Generation from the first episode, as the first season sets up characters and relationships referred to throughout the series. There are 7 seasons (the last two are concurrent with the start of DS9), so you may not get through all of these, since you're not interested in watching everything. However, if you do end up sufficiently captivated by the universe, then you'll be in a good place to continue - many of the relationships and conflicts in DS9 and Voyager are originally set up in TNG.
Enterprise has mixed reviews (I personally felt that some parts were very good) - since it's a prequel, it's also the easiest to skip. Placing this last in viewing order means that you don't miss anything significant if you're tired of Star Trek by the time you get here, but can continue in a satisfactory order if you're not.
If you get to the end of season five of TNG and aren't yet decided whether you want to put much more time into the universe, then I'd suggest continuing with TNG and then Voyager, and skipping DS9 (potentially returning to it later). After the earliest episodes of Voyager, the DS9 history isn't particularly important, since it takes place far away, and Voyager is a stronger show in terms of "ships in space" (also in general, IMO).