Shonky,
This is a pretty difficult question because Rush has had (in my opinion) as least 4 really different sounds since they have been around (and even at that a lot of these sounds changed a lot over their albums). If you want a good overview of the band, you might want to get Chronicles, which is a two-disc set and covers their stuff from the seventies and eighties.

There are two problems with this set however:1) It does not cover their newest stuff which you may be interested in if all you have ever heard is the teaser and 2) one of the bands great strengths is to create albums with a great deal of synergy which is lost on a compilation type album. It is, however, a great value since they put 74 min worth of music on each of the two discs.

If this teaser is all you have ever heard, then you are probably interested in their newest sound, which is quite different from their sound in the eighties and seventies. If you like edgier stuff I would try Test for Echo and Counterparts. This is by no means the best of Rush, but it is the closest to what you are hearing in the sample.

The three albums you mentioned are actually some of the band’s older work, but have withstood the test of time. 2112 was in 76 and their first concept album. The "A Side" of the disc is one twenty-minute song that tells a story set in the future. This is a landmark album for Rush as the next two albums also featured concept type songs.

Moving Pictures was recorded in 1981 (I think) and is the album I most often hear people saying is Rush's best. If you want to know Rush, this is my highest recommendation for a first buy. It would be very difficult to find someone who enjoys the music of Rush and doesn't care for this one. The "A Side" of Moving Pictures has four songs and each of these still receive regular air play on my local Rock station (Ok, well they did until I got my Empeg. I have no idea what they play now). The "B side" has three songs, all three of which are very good. "Tom Sawyer" is on this album which is by far the most recognized Rush song. This album is the point they started really adding keyboards heavily into their music.

Later in the eighties Rush really got into keyboards and I think this is the point some fans got disappointed. I (and Tony as well) really like the keyboard stuff, but it sounds very different from the 2112 sound. Of this eighties stuff Tony really likes Power Windows, but I will let him sell you on that one. I like Signals and Hold Your Fire the best, but I don't think I would recommend these as first buys.

After this, the band got into a progressive sound I really don't know how to describe which really (in my opinion) only lasted two albums - Presto and Roll the Bones. The band laid off the keyboards here and really started emphasizing a clean technical sound. Roll The Bones is my favorite Rush album, but I think I am the only one who thinks that so this is probably not a good first buy either.

Counterparts was the next album and saw the band get a little heavier. It wasn't totally different from the previous albums, but it was a lot closer to Test For Echo than anything else. Test for Echo sounds (to me) a lot like the clip from the new album and was a very different sound for Rush. Though this is a great album, I really wouldn't recommend it as a first buy unless all you have heard is the newest stuff and that's what you want to buy.

Sorry to be so long winded but it’s difficult to give recommendations about Rush. I’d wait to hear from Tony before you commit any dollars, but I figured you might want to hear from more than one source because this really is a very prolific band.