In the film Prometheus, is the planet shown in the beginning of the film Earth or was it some other planet?
The planet's location is deliberately undetermined.
From this movies.com interview with Ridley Scott:
Movies.com: That is our planet, right?
RS: No, it doesn’t have to be. That could be anywhere. That could be a planet anywhere. All he’s doing is acting as a gardener in space.
So it could be Earth, but it might not be, and it doesn't matter to Scott or the story; that scene exists as a generic representative of a process which occurred on an unknown-but-large number of worlds (including Earth) in much the same way.
(The scene depicts a real-life Earth waterfall, but that shouldn't be taken as meaning it's set on Earth--the crust of the Earth at the time the scene takes place has been recycled through the Earth's mantle since then, so the choice of waterfall is purely aesthetic.)
What Andrew Thompson said. It is acknowledged to be a primordial Earth in the screenplay - even the far superior, pre-Lindelof version - and the film itself heavily, heavily implies it, though it is never explicitly stated. I'm surprised you would consider it to be another planet for even a second, to be honest.
Here's the original Jon Spaihts screenplay (Prometheus was still called Alien: Engineers then). And this is the very first line in said screenplay:
EXT. EARTH - DAY (12,000 B.C.)
Now, continuing further, as if that wasn't enough.
The world turns below us, vast and slow.
A RUMBLE. A shadow sweeps over the land. We move with the shadow. We cast the shadow.
Landscapes slide by. Reduced by altitude to abstractions: river deltas, forests and flood plains. A raw natural world. No trace of civilization.
The shadow glides over mountains and glaciers. Across an ocean and a pale beach.
Over lowland plain at the foot of a VOLCANIC MOUNTAIN it stops.
EXT. LOWLAND PLAIN - DAY
THREE FIGURES walk out of the shadow.
They are men - and yet not men. Their skin is snow-white. Their features heavy and classical - as if Rodin’s Thinker had risen from his seat. Their smooth heads are earless and hairless. Their glittering eyes entirely black.
Against the stark land their height is impossible to judge. They are ENGINEERS.
Two of them are cloaked in dark robes of strange design.
The third is naked.
One of the cloaked Engineers opens a featureless black box: inside lies a cake of dark, sticky material.
The naked one lifts the dark cake with ceremonial slowness. It hums and buzzes. Foams into iridescent spheres. He raises the seething cake to his mouth like the sacrament.
BLACK SCARABS boil out of the dark material. Swarm over his lips. Glittering insects that chitter and bite.
Under the swarm his lips melt away. A horrific vision of teeth, black blood, dissolving bone. They are devouring him.
I don't think I need to go further. While there were some changes in the finished film, such as there only being one Engineer, this seems to be fairly clear evidence that the opening scene was always intended to be on Earth. Even Lindelof's inferior screenplay agrees with this:
In the beginning, there was...
EXT. EARTH - DAY
SOUNDS. A GRADUAL RUMBLE -- SOMETHING MECHANICAL -- AN ENGINE? And then, SUDDENLY --
BRIGHTNESS. So sudden it HURTS our eyes... but now it gives way to --
WATER. Intensely BLUE, untouched and PRISTINE as the rays of the SUN dance off its glasslike surface. CLEAR. UNTOUCHED. And we’re SOARING OVER IT -- The RUMBLE -- DEEP AND LOW -- And now we SEE SOMETHING moving over the clear water --
A SHADOW. Shaped like an enormous HORSESHOE. And whatever we are IN right now, THAT is what’s casting it. MOVING ALONG THE SURFACE at increasing VELOCITY as we finally hit --
Again, I could go on, but this answer is long enough. It's obvious - even though this is, admittedly, not Lindelof's final draft, as I couldn't find that - that the planet was always meant to be Earth, Ridley Scott's ex post facto comments aside. Enjoy comparing the screenplays I've linked, fellow masochists.