i think jimi's 1st album is pretty brilliant.

you can completely ignore electric ladyland, and albums one and two still stand.

hmm, not easy, i'd go for Emperor - In the Nightside Eclipse and Anthems to the Welkin at Dusk and Morbid Angel - Altars of Madness and Blessed are the Sick...

Rush - Rush & Fly By Night
Heart...anything....
Yes...anything
Kansas This list could be quite extensive...

looking for best 1st album release, and then followup...

i'd say heart's second album was not as strong as the first...
and rush's 1st album was hardly ground breaking,
yes could possibly in the running, 1st album was great, 2nd was even better......
kansas?
hm.

jury's out on I, tho song for america was REALLY strong

batsbrew


i think jimi's 1st album is pretty brilliant.

you can completely ignore electric ladyland, and albums one and two still stand.

Not trying to give ya hard time...but it doesn't fit the criteria....not when the 3rd album
is better...at least I thought the OP wanted the bands that after their first two- couldn't hang.
 
At least you have an entry...the only band I could think of was Free "Fire and Water" but it was
their 3rd album..sheesh.
 

 
+1 to Iron Maiden: Iron Maiden/Killers - Paul Di'Anno was a much better singer for Maiden than Brucie. I'd argue that Blaze Bayley was better than Bruce too. 
 
I'll go with:

• Motörhead: Motörhead/Overkill
• Rainbow: Rainbow/Rainbow Rising
• Stiff Little Fingers: Inflammable Material/Nobody's Heroes
• The Stranglers: Rattus Norvegicus/No More Heroes

Zappa

Grand Funk Railroad

The BAND

Thought I'd weigh in on this one. First 2 Bad Co. albums (Bad Co. and Straight Shooter). I don't think any album of theirs after that was worth the cost of the vinyl.

Well, that certainly brings FREE into the mix....

J

Ambrosia:  Ambrosia & Somewhere I've Never Traveled.