It seems to me that the addition of electrons and protons as you move across a period would cause an atom to become larger. However, I'm told it gets smaller. Why is this?
As you move from left to right across a period, the number of protons in the nucleus increases. The electrons are thus attracted to the nucleus more strongly, and the atomic radius is smaller (this attraction is much stronger than the relatively weak repulsion between electrons).
As you move down a column, there are more protons, but there are also more complete energy levels below the valence electrons. These lower energy levels shield the valence electrons from the attractive effects of the atom's nucleus, so the atomic radius gets larger.
Remember, the 'size' of an atom has nothing to do with the size of the nucleus. It has to do with the size of the valence shell (which itself is not well-defined*).
So, if we neglect change in electrical attraction, the size should stay the same—a shell is a shell and it need not 'expand' to accomodate electrons.
Now, as we add more protons and electrons, the attraction between the nucleus and shell increases and the shell contracts. Thus the atom gets smaller.
*Shells reach 'til infinity, so it's better to define the size on the basis of 'the electrons of the outermost shell will be within this region x% of the time.' This only changes this answer a tiny bit. Now, we say 'the probability of finding the electrons closer to the center increases due to increased nuclear charge.'