There seems to be a constant debate between which recording of Assassins people prefer--I personally think (despite some very good performances in the Broadway), that the Off-Broadway blows it away in terms of overall atmosphere (I admit, I'm not a fan of the added Something Just Broke).
I guess you really should start with Sunday, and the original recording is near perfection (and I'm not even a Mandy fan), although I admit that's one show that I probably put on the DVD of at least as much as the CD (but, I'll assume you don't own the DVD either, so that may be a moot point).
The OBC of Whistle is just thrilling (even the quickness it was recorded in, with several flubs, kinda adds to the theatricality for me, much as it does with Merrily's OBCR). The concert is pretty expensive to buy now, anyway, but I'd choose the OBCR anyday.
Pacific Overtures and Passion may be my two fave scores of his (not necesarily shows, though I enjoy both a lot). Simply gorgeous. The revival recording of PO has a tiny sounding orchestra and lacks, for me, theatricality, and I'm not a fan of the English National Opera one (though the out of print two disc is complete, dialogue and all), but, even with cuts, the original cast sounds superb (frustratingly, maybe because it's oneof his least done shows, it's the only pre 1990 Sondheim original cast album yet to be remastered, though it's still in print--when they did the RCA Sondheim remasters they quietly just ignored it, but the CD sounds pretty good, regardless).
Passion needs to start with the OBCR as well (actually for Sondheim, nearly always, the first cxast album is the one to stick with).
I agree that Frogs, Road Show/Bounce and Saturday Night can go after those. That said, I think Saturday Night is an absolutely winning, and charming show--it's almost as if Sondheim was doing (and doing very well) a musical comedy in that 1950's style of Wonderful Town/Pajama Game, etc, and I suppose he was. In this case, do not get the first recording, from England (it may not even be in print)--the later Off Broadway recording from Kathleen Marshall's production is a huge charmer, with great performances, a fuller score, and proper Tunick orchestrations. A few years ago, I found it from Amazon Marketplace new for about 3 dollars--it had a "cut" in the jewel box of the CD meaning it was a remainder, so maybe they printed more editions than they came close to selling and more cheap remainder editions are out there. Anyway, I think the score is underated, and it would be next choice.
Road Show/Bounce is worth listening to after you have those, and a lot of it has grown on me, but it's hard to recommend before the others. Frogs, as well. Actually, I like the choral music done for the original 70s Frogs very much--I don't much care for the added bits, and there is a recording of the shorter, 1970s score combined with a good recording of the songs of Evening Primrose (if you don't already have that soundtrack, which while a limited edition Kritzerland still seems to have copies of), with Neil Patrick Harris that makes that edition the one to buy first, IMHO.
I believe all of those recordings are in print (if for some godawful reasonyou ignore Gaveston and myself,

, and choose the revival of Assassins, that was out of print for a while but PS Classics just released more copies). And like I said, you can often find cheap, and still new copies of them from Amazon Marketplace which I know amazon gift cards will allow yout o purchase through.