What you most likely want is an Ellis wormhole. The picture in A. Forty's answer most likely shows exactly that; and as others have mentioned, Interstellar is about as good a visualization a wormhole as you're going to get from Hollywood. However, I don't think Interstellar got all the details right.
From the outside, the wormhole would look very much like A. Forty's image. Moving through the wormhole, the image of the other side of the hole would appear to expand around you, swallowing you up, and then you're on the other side.
If you were to look left or right (or up or down, for that matter) while in the middle of the wormhole, you would see the back of your own head. The hole has no walls; each side just wraps around to the opposite side. I believe Interstellar goofed on this one.
Shown here is what an Ellis wormhole in a 2-dimensional universe might look like from the outside:
Interstellar showed something like what you would see if you piloted a spaceship along this surface and through the hole in the middle, from one side to the other. However, in reality, the wormhole would be three-dimensional. You'd never actually see anything even remotely resembling the above surface- because we're inside it.
Imagine taking a 2-dimensional spaceship and sliding it around on that surface. Try to visualize what the ship's pilot would see, looking at the wormhole. Near the hole, his lines of sight would be bent toward it, so objects behind the hole would appear distorted and stretched out at the edges of the hole. Closer to the hole, rays of light could wrap around the hole multiple times before coming back out without going through, so the pilot would see all 360° of his surroundings (including the front of his own ship) reflected back and forth, squeezed into tighter and tighter rings.
Just inside that, an infinitesimally thin black ring coinciding with the black ring in the image, where light (and anything else) can zip around and around forever.
And inside that, he'd see into the world on the other side of the wormhole.
In the center of the wormhole, his line of sight would fold back around, like the black line on the right-hand side of the image. Off-center, the space would bend the pilot's line of sight at a much larger angle, so he'd get a (very distorted) 360° view of the other universe. The effect would closely resemble an image taken with a strong fish-eye lens, like the ones used in those 360° cameras. As before, all of those 360° would be repeated over and over, compressed into tighter and tighter rings approaching the black ring representing the halfway point.
Now compare all that to A. Forty's image.
Now, imagine that the pilot takes his ship through the wormhole. What does he see?
Well, first thing, as he approached, the fish-eye view of the other side would start occupying more and more of his field of view, as would the distorted space around the hole. Upon reaching the halfway point, light emitted from the left side of the ship would circle around the hole to the ship's right side. So by looking out either window, he's just see the outside of the ship from the opposite side. The black ring is now whatever color his ship is. Watching out the back window while exiting the hole would look much the same as going in did, except that the pilot's original universe now fills the fish-eye view in the center.
In three dimensions, the wormhole experience would be fairly similar, except the halfway point would be a spherical surface (which would appear as an infinitesimally thin black ring) around which light can orbit.
If these wormholes somehow exist in a gravitational field (because they're on Earth's surface, say), that should still work without any completely broken physics or anything. I'm not completely sure how it would work, but I have a guess.
Say the people on one side of one of these wormholes decide to build a bridge through it, so they can just walk through, from one side to the other, without having to jump through it or anything, because that sounds hard. This should be doable; there might even be normal gravity all the way through. If you were to fall off the bridge inside the wormhole, you'd just drop out of the hole on one side or the other. If you fell off the exact center of the bridge, you might wind up in a sort of gravitational saddle point. You'd feel as if you were being stretched in one direction (along the X axis, call it, toward the mouths of the wormhole), but gently compressed in the other two (Y and Z). You could float there as long as you were perfectly balanced in the X direction, but just a little bit off and you'd fall straight out the bottom of one side of the wormhole.
And finally, as to what being in one of these wormholes might feel like. Note how, in the image above, all of the space in and around the hole is curved kind of like a saddle or a Pringles chip, so if you drew two intersecting "straight" lines on the surface, they could curve in opposite directions. For instance, the two black lines shown. The center ring curves toward the hole in the center, while the line on the right curves away. This means the space has negative Gaussian curvature. A funny thing about negative-curvature surfaces is that there is less space inside any shape drawn on the surface than you'd expect, based on its perimeter. This will do... weird things... to anything put into the wormhole.
Imagine you're sliding a small square of paper around on the surface shown above. If you try to slide it to the hole, you'll find that it won't stay flat on the surface. Either the center of the paper will pop up above the surface, or it'll just kind of crinkle. If you wanted it to lay flat, you could cut a notch from the edge of the square to the center, so it could spread out a bit and sit flatter on the surface.
Now... what would you happen if you put your hand into one of these wormholes? Quite suddenly, there would no longer be enough space inside your hand for all the bone and muscle and such that is there. Your skin would suddenly feel very tight, and some of the blood in your hand would be squeezed back into your arm. The bones in your fingers and hand probably wouldn't take kindly to being distorted in this manner, either. Like the piece of paper, they might just crack.
Best make these wormholes quite large, so these effects aren't strong enough to cause harm to something the size of a human.