Woodwork is a hobby of mine and oil is my favourite finish. I would suggest going for a drying oil. Don't worry about the type as there is as much difference between the various makers as there is between the types. My personal favourite is Rustins Danish Oil.
To apply the oil, just spread it on with a cloth or paper towel. Leave it alone for ten minutes and wipe off everything that has not soaked in. Leave it for a day to dry and then repeat. The more thin coats you put on, the shinier it becomes. Whatever you do, though, don't scrunch up the cloth/paper towel and leave it. As the oil dries, it can create much heat and even spontaneously combust. Take the cloth outside (or pun in wood burner) and set fire to it. It will just flare up and then pose no further risk. When I have put on the last coat, I like to give just one coat of Briwax Original. This takes off the oily feel and also gives it a nice smell.