Oxidation and reduction aren't two separate reactions: they are 1 reaction, called a redox reaction. So, like in every chemical reaction, the atoms of different elements/compounds rearrange themselves and form new elements compounds.
This is done mainly by swapping electrons. Oxidation and reduction attempts to explain to us what really is going on when these compounds form and tries to justify why certain compounds were formed (and not others). E.g. in the dry cell:
The reason Zinc hydroxide and magnesium hydroxide are formed is because we know from the oxidation half equation at the anode that Zn reacts with the hydroxide ions to form zinc hydroxide, and releases 2 electrons in the process.
Now these electrons flow to the cathode, where they react with water and Magnesium (just realised there is an error in the photo, it is meant to be Mg not Mn) to form Magnesium hydroxide. The reason the electrons flow towards these compounds is because the reaction at the cathode is "reducing" (aka absorbing) all the electrons in the whole system due to the chemical reaction at the cathode, and so can be thought of as having a positive charge. Electrons have a negative charge and so are attracted towards anything positive. Overall, you can just write this reaction as one equation (refer to third equation in photo).
Although, as a general rule, metals tend to oxidise and donate electrons and non-metals tend to reduce and absorb electrons, this isn't always the case. But thank BOSTES, they provide you with the table of standard potentials present at the back of the periodic table they provide in the HSC (Hence Sien's suggestion). This table lists all the oxidation/reduction reactions relevant to the HSC course, so you won't have to memorise which elements oxidise and which reduce given a certain redox system.