You can observe atoms using a variety of techniques including transmission electron microscopy, and scanning tunneling microscopy. In transmission electron microscopy a focused beam of electrons is passed through a thin piece of material. By looking at how many electrons make it through as you shine the beam at different parts of the sample, you can map out the structure of the material. If the material is regular (like a crystal) and very thin, you can even start to see atoms. Here's a picture I found on the web:


http://www.research.ibm.com/journal/rd/444/batso12.jpg


You can also see atoms using scanning tunneling microscopy. In this case you monitor the current that flows between a very sharp metallic tip as it is scanned over a conducting surface. Take a look at this picture for an example:


http://www.aip.org/physnews/graphics/html/corral.htm


Each of the blue spikes in this image is a single iron atom on a copper surface that the users of the scanning tunneling microscope have carefully pushed into place using the STM tip. The copper surface (colored red in the picture) appears to have a regular pattern of waves because the electrons on the surface are confined to wavelike patterns.


For more information try searching the web for articles about TEM, STM, AFM or XRAY crystallography.

Some molecules ( DNA for example ) are actually big enough so you can look at them under an electron microscope. Others are not quite as big but you can attach a metal tag to them and then see them under an electron microscope. If the molecules are still smaller you can shoot X-rays through them and observe how molecules of different shapes deflect the X-rays in different manner.
For very small molecules scientists look at a piece of a compound and do various experiments on it and the best way to explain the results of the experiments is thatthe compound is made of many tiny molecules which are all the same and add up to give the properties that you observe for the substance as a whole.

This is a very interesting question on several levels. Traditionally, the concept of atoms and molecules has basically been proven by having developed a model for what atoms and molecules are, and then doing many different types of experiments to see if the results are all consistent for our model of the atom. For example, by hitting a crystal (a periodic structure of atoms) with electrons or X-rays, we obtain diffraction patterns can be nicely predicted by our models involving atoms. But you may still ask if it is the correct model because we can't "see" atoms. As it turns out, in the early 1980's, a new analytical technique was invented called "scanning tunneling microscopy." This technique is much different than optical microscopy, but instead involves transferring electrons back and forth from a surface to a probe tip.


I won't go into details, but imagine having a very sharp pin, and slowly moving it across a flat surface. By applying a bias between the pin and surface, there is a net flow of electrons between the two. By keeping that flow of electrons constant, we can effectively map the electronic density of the surface. To picture what that really means, try making a model of row of atoms, like a bunch of balls (marbles, tennis balls, whatever...) in a row where each ball represents an atom. Now, let's think of the curvature of the balls as where the electrons are, such that the curvature represents the "electronic density" I had mentioned earlier. Now, if you take a pencil or some some of sharp object, and move it along the row while maintaining a constant separation between between the tip and the balls, you'll basically map out the surface showing the curvature. Since each ball was an atoms, what we see, is that this technique allows us to "see" the atoms (or, more specifically, the electronic nature of the surface.).


There is also a similar technique called "atomic force microscopy," that is similar to scanning tunneling microscopy. The concept is basically the same, except now we imagine that the tip is effectively "touching" the surface and moving along the row of atoms. Here, instead of monitoring the flow of electrons, we are relying on atomic forces (and repulsive forces) to allow us to map out where the atoms are.


Proving that atoms exist beyond these techniques (and the previous consistency checks with our models) may ultimately be more of a philosophical issue. If one only believes in what they can "see," then at some level, they will also have to define what is "seeing." Right now, we have a model for atoms and molecules that fits all our experimental observations to date. Does this mean we're right? Maybe, but maybe not. There are many examples throughout the years where refinements are constantly being made to ideas we once thought were "final," including many changes to our picture of what an atom is. You're asking the right questions, and you shouldn't take it for granted what is accepted as "common knowledge." Don't forget, at one point everyone thought the world was flat and the earth was the center of the universe!


If you would like to learn more about imaging atoms and what you see, feel free to get in touch with me - I did the majority of my graduate work using these techniques and would be happy to explain them in more detail to you.