Yeah, that's pretty much a no-frills version of it.
I think that anyone who likes Maiden will have no big problems getting into Amon Amarth. By death metal standards, they're easy listening, very riff-heavy and melodic. They will probably make you start to appreciate growls and harsh vocals, too. Maybe add Children of Bodom to that list. They're not death metal, but they might help to nudge you to that direction.
God Dethroned is also easily digestible for a heavy metal fan. They're quite catchy and melodic, but still none the less very much death metal. Start out with their recent masterpiece,
Passiondale or their even better
Lair of the White Worm album.
Oh, and listen to Slayer. Slayer aren't death metal either, but they will help get you access to the genre.
As for black metal, if you're totally unsure, maybe Primordial is a good place to start. They have clear vocals, but very black-ish guitars and arrangements (combined with folk elements). Just take a good listen to their 2007 masterpiece,
To the Nameless Dead. I've never known an album in recent years to be so unanimously celebrated as this one... and it lives up to its hype.
Vreid, as I mentioned, is a good starting point, too. They are quite accessible, and simply extremely good.
Milorg (2009) is pretty much the best album released last year, in my opinion.
If you want some more clicheed but still good and fairly accessible black metal, check out Dark Funeral. They cover pretty much the entire spectrum of what black metal has to offer and should help anyone appreciate the genre.
Note, these are all personal suggestions. Other people may have both different opinions on the bands I mentioned, or may suggest entirely different band for starters. Whether you take their advice or mine, try concentrating on these bands first. Get familiar with them and see if you start liking them. If no, don't bother to go further. There is plenty of other metal out there. If yes, check out the band's backgrounds and start exploring.
Some good samples:
Amon Amarth
Pursuit of Vikings
The Hero
God Dethroned
Sigma Enigma
Passiondale
Primordial
Coffin Ships
Empire Falls
Vreid
I Krig
Speak Goddamnit
Dark Funeral
An Apprentice of Satan
Atrum Regina
For the brave, some bonus samples:
Marduk - Fistfucking God's Planet
Bloodbath - Eaten
Asphyx - Eisenbahnmörser
And finally, one of the best live covers I know:
Vader - Raining Blood