The reactivities of elements can be predicted by periodic trends (patterns within the Periodic Table.)
Let's start with periodic trends. An element's reactivity is its tendency to lose or gain electrons.
The Alkali Metals (Group 1) have the highest tendency to lose electrons, making them the most reactive of the metals. This is because they have one outer electron and losing it gives them the stability of a outer electron shell as the next level down becomes the highest occupied level. The Alkali Metals become more reactive moving down in the group, with francium being the most reactive. This trend occurs because each element, as you move down, has more inner electrons that shield the one valence electron from the nuclear charge making it less attracted.
The Alkaline Earth Metals (Group 2) are the next most reactive of the metals. They achieve stability by losing two electrons. It takes more energy to remove a second electron, which is why they're less reactive than the Group 1 metals. The trend for metals is that they become more reactive going down and less reactive going across in the periodic table.
The Halogens (Group 17 or VIIA) are the most reactive non-metals because they have the highest tendency to GAIN an electron. This is an extension of the periodic trend that the energy required to remove an electron increases going across a row of the periodic table. At a point it becomes more energetically favorable for an atom to gain electrons than to lose electrons. The Halogens, having 7 valence electrons, achieve a full outer level by gaining one electron. The elements of the Oxygen family achieve a full level by gaining two electrons. The trend for non-metals (Groups 14-18 or IVA - VIIIA) is that they become more reactive moving up within a group and moving to the right across a row.
The noble gases, Group 18 on the far right, are the least reactive of all elements because they have a full outer level of electrons.
There is published data to help you determine relative reactivity of the transition metals, which don't follow trends as closely. The activity series of metals is a ranking of order of reactivity.
It's also possible to create one's own activity series by reacting several elements with the same compound, for example reacting a set of metals with the same acid and ranking them by observation of rates of reaction.