From your description, it sounds like you have bought an old pine Welsh Dresser. It probably has just a wax finish on it, which is great but and there are 2 buts, and they are:
a. The piece has probably been polished to death over the years with Pledge etc and this is not good as it will now be covered in silicon which will react/interfere with any finish you decide to use. So you probably want to consider just stripping it to bare wood anyway unless you want to keep any patina that has built up, then you need a light sanding 400g over the whole piece and a clean done with some acetone damped rags.
b. wax can look great and be as shinny as you want, depending on how much elbow grease you are prepared to put in and it is easily repairable. Water ring are no problem at all, just a splash of lighter fluid over the top of it and light it - give it a few seconds and blow it out when the ring disappears. Let it cool a little and apply fresh wax. The But is remember it is not a waterproof finish merely repellent and does not like hot things on it at all.
If you go the route of putting on an oil finish then you will have to go back to bare wood (as you don't really know what is on it) starting with 120g and then working up to 320g not skipping any grits in between as each is needed to remove the scratches from the previous lower grit. Then you need to flash off with some acetone to get rid of the larger dust particles without raising the grain and then go over the whole piece with a tack rag. Once done apply your oil. For the best results you should follow the old maxim about the application rate for a new piece "Once a day for a week, once a week for a month and then once a month for a year." When you put it on leave it for around 20 to 30 mins (surrounding temp dependant) and wipe off.
DO NOT PUT OILY RAGS IN BIN - lay them out flat outside to dry or hang on washing line until dry. There is a high risk of the oily rags spontaneously combusting and burning the house down.