Dio: I agree about Dream Evil and Magica. They are excellent, and the song "All the Fools Sailed Away" on Dream Evil is criminally underrated. Magica is a concept album with a totally lame story, but the music is as strong top-to-bottom as any Dio album. Master of the Moon was Dio's last album, and has some good songs, but some filler. The other albums you can skip, but there are a number of anthologies and greatest hits albums that for the most part pick up the highlights. EDIT: In the U.S., Magica is now sold as a 2-CD package with Killing the Dragon for about the same price as a single CD, so that's a pretty good value for your entertainment dollar. I listen to Magica far, far more often than Dragon though.
Black Sabbath: Master of Reality, Heaven and Hell and Mob Rules are essentials. If you like Master of Reality, then check out Volume 4 and Sabbath Bloody Sabbath. If you like the Dio-fronted albums, then also worth checking out Dehumanizer, which was released in the early 1990s and also featured the Dio/Iommi/Butler/Appice lineup. Dehumanizer isn't quite as strong as Heaven and Hell or Mob Rules, but it does feature the song "I" which is completely awesome.
Rainbow: You could cheat (like I did) and get the Catch the Rainbow anthology, which does a pretty good job of covering the best Rainbow songs. Disc 1 covers the three Dio-fronted albums, Disc 2 covers the albums with Graham Bonnet and Joe Lynn Turner singing. I'm a much bigger fan of the Dio material on Disc 1. Dio recorded two other studio albums with Rainbow, Ritchie Blackmore's Rainbow and Long Live Rock 'n' Roll, neither of which is as strong as Rising, but both are classics. The Live in Munich concert CD/DVD is not a bad supplement to Rising, if you can handle Blackmore's extended solo jams, because it pretty much just covers material from the first and third albums, with only one song from Rising.
Can't really speak to Diamond Head or Angel Witch, though I did pick up the 25th Anniversary edition of the latter, which is fun because it includes all the scathing reviews Angel Witch got in the English press at the time. I think it's good but not great, and I'm in no hurry to pick up other Angel Witch material.
EDIT: Perun once again typed faster than me, but we're largely saying the same thing.