Valence Shell Electrons
It's often important to know the number of valence electrons -- electrons available to form bonds. You can figure out how many valence electrons an element has by counting columns starting from the left. If you're looking at sodium, for example, you would deduce it has one valence electron because it's in the leftmost column. Chromium, by contrast, is in column 6. To reach it starting from the left you have to count six columns across, so chromium has 6 valence electrons. If the element is in the p-block (groups 13-17), disregard the d-block (groups 3-12) when counting columns. If you're looking at arsenic, for example, you would start counting from column 1 and skip the d-block, the columns from scandium through zinc. Counting in this way will give you a total of five columns over from the left, implying that arsenic has 5 valence electrons.
Electronegativity & Size
As you cross the table from left to right or bottom to top, the electronegativity or electron-pulling power of the elements tends to increase. The LEAST electronegative elements are in the lower-left hand corner of the table, while the MOST electronegative elements are in the upper-right hand corner. In comparing fluorine and selenium, for example, you would deduce that fluorine is more electronegative, since it lies farther up and to the right. The size or atomic radius of the elements, on the other hand, behaves in exactly the opposite manner. As you go across the table from left to right or bottom to top, the elements tend to grow successively smaller. Iodine, for instance, lies near the bottom of a column and has an atomic radius of 140 picometers (trillionths of a meter), while fluorine at the top of the same column has an atomic radius of 60 picometers -- less than half the size.
Nucleophilicity
Nucleophilicity measures a chemical species' tendency to donate electron pairs to electron-poor molecules or groups. Here there are two different trends. As you go down a column, nucleophilicity increases. Iodine, for example, is a better nucleophile than fluorine, while sulfur is a better nucleophile than oxygen. In the rows, by contrast, nucleophilicity increases as you go from right to left. A negatively charged carbon atom, for instance, is a much better nucleophile than a negatively charged fluorine atom. It's important to note, however, that an atom or group with an overt negative charge will generally be a better nucleophile than an atom or group with no net charge.
Polarizability and Basicity
Polarizability measures the ease with which the electron cloud around an atom can be distorted by interactions with other atoms or molecules. Polarizability is strongly correlated with size, so it tends to increase as you go right to left and top to bottom. Basicity measures the tendency of an atom or molecule to pick up hydrogen ions. Here the trend is a little bit different. Basicity increases going right to left and bottom to top. Fluorine, for example, is much more basic than bromine but much less so than carbon. This trend explains why hydrochloric and hydrobromic acid are much more acidic than hydrofluoric acid while methane is not acidic at all.