I like Lemon Oil, it smells nice.... also you can get food safe danish oil also smells nice, darkens the wood a little, works great with some grain patterns, or as above sunflower oil is cheap and adds some waterproofing. Reality is a half liter tin will last ages so you forget the price before you need a new one.
Or get some bees wax and polish it with that, after all once it's out of the workshop it will probably only ever see wax furniture polish for the rest of it's life!
Down the line a bit you will probably want some sanding sealer, this is cellulose based, you want it quite diluted which means it lasts a long time, so you will want some cellulose thinners... you can buy that by the gallon at car paint shops for the price of a small tin at halfords or wood shops.
First thing I polished was a lidded box, I used motorcycle wax to polish it... still looks good today. Play and pick up different finishes as and when and find what you like, get small quantities a dab of oil or wax on a cloth goes a long way when the lathe is spinning... I suspect in fact I know cos I used some the other week, I still have some stick wax that came half used when I bought the lathe in 2009.