(I'm totally going off the top of my head. Anyone please correct me if I'm wrong.)
The Master System and Genesis games Phantasy Star 1,2 and 4 are definitely connected and take place in the same star system. I can't vouch for 3 as I've never got into it, but I believe it is also connected. There is also a Game Gear game called Phantasy Star Gaiden which I believe is connected as well.
Phantasy Star Online series is a completely different story-line and setting(universe?), but has similar themes.
The DS series Phantasy Star Zero is connected to Phantasy Star Online (I believe) as is the vita game Phantasy Star Nova.
Phantasy Star Universe again is a completely different story-line from the others.
The PSP Series Phantasy Star Portable is connected to Phantasy Star Universe
Basically, the order would depend on what you're interested. If you want traditional 8bit/16bit RPG experience, I would do Phantasy Star 1, 2 and 4 (did anybody like 3 at all?). Especially Phantasy Star 4, which tries to cover all the themes and plot lines of the previous games before it, but with the most updated combat, sound and graphics. If you have to pick one from the all these, do 4 and you'll be good.
If you want an very sparse, loose storytelling but excellent atmosphere and online play. Go with the Phantasy Star Online series. Track down the PC version and see if you can find a community server to connect to. There was a popular one called schthack but I think it has closed down. Or jump into Phantasy Star Online 2 (via an English Patch).
People rag on the Universe series, but I enjoyed it. It was an MMO like the Online series, but they did add a more structured single player storyline. Track down an Xbox copy to play as I don't think the PC version will work anymore.
Phantasy Star is a classic series and each first version of the game was groundbreaking for its time. '
To answer you're question about the Sega Saturn. It only had compilations of the MS/Gen versions I believe. There are remakes of the first two that published on PS2. English versions of them are available on the net.
Hope this helps.