Contra III - levels 3 and 4. Seeing this through a five year old's eyes, I'll never be filled with this wonder again.
Ghouls 'n Ghosts - level 1.
Strider - level 2. From Siberian wilderness to Roboape to I GOT YOU guy to power station and eventually Balrog.
Mega Man - Elec Man stage. Shame there weren't more vertical scrolling levels in platformers.
Super Castlevania IV - first level, always replayable. When I first played it through component on the Wii VC it was crazy how much detail I had missed before.
Streets of Rage 2 - elevator stage, legendary.
Final Fight - level 2, subway. Sometimes I wouldn't move so far just seeing how cool the Mad Gear guys looked, sitting and waiting for me to show up.
Metal Slug X - train stage.
Gradius V - level 2, working side-by-side with yourself from the future.
Just some off the top of my head. Surely there must be more. I didn't say anything from Irem, or Treasure, or others.
Capcom always had strong opening levels, that's for sure.