I would check the Ellis Island online website at: http://www.ellisisland.org/

There are passenger lists for free, and it will find the ship your ancestors came in on. Last names (surnames) really don't say too much, so don't take too much stock in the name.

Have your father write down your grandfather's name, date of birth/death, all known marriages and children, and where his father was born and all other pertinent information. Make sure you write it down.

Finally--cherish the stories of your family history. Stories are lovely things that exist as a part of our family fabric. There are always little gold nuggets in the stories that are there to guide you. Just make sure when you do genealogy research that you base research off of known facts first. :)

I do want to warn you that sometimes family intentionally conceal their past (a billion reasons). This may not be your case, but sometimes you will get resistance if that is the case.
Good luck.