Hi, Kate. You can use many methods to finish furniture. The choice depends on how you want it to look and how much protection to the surface you need. Alder is a relatively soft hardwood. It will be more easily subject to damage from bumping into another piece of furniture or a dropped dinner knife or something.
Since you don't have much experience in this area, I suggest that you get a book called "Understanding Wood Finishing" by Bob Flexner. It is available from all the major bookstores and on line. It also contains a lot of other good info on keeping your other furniture in good shape and it is written in a very easy, readable style. My copy was $16.95 a few years ago. There is nothing about alder wood in particular in it though since alder has not been a common furniture wood.