Yes, there are thousands of free sites where you can find your history. Here is the catch.. expect to do it as a puzzle, not a finished package, and you put together the pieces. The other thing to bear in mind is that info about living persons is not usually online. If your father is not living.. then the first record you need is his death certificate, and his birth certificate. These should show the parent names.
Once you reach someone born before 1930, they are LIKELY to be in the census records which go back to 1790. Ancestry offers the convenience of looking from home at these .. or your local library probably subscribes for your benefit. You may find that the ability to do this from home, without paying a baby sitter, winds up actually being cheaper.
Notice that my focus is RECORDS, and not websites. Not all genealogy is online. That is just reality. You can stop at www.rootsweb.com (which is free) and has a tutorial on the process of building your tree, and what records you need to find to do this. It all depends on the specifics. You have 8 gr grandparents.. it is possible that of those 8, you will bump into someone who already has researched it back (as Shirley explains..this is not always reliable), and 16 gr gr grandparents doubles the odds.
There is no one best site, because they all come from different places and times. The MORE sites you use.. the better the odds that you can pick up a death date here, a marriage record there, a probate file somewhere else.
95% of my research is without using ancestry.com. I can't begin to tell you the amount of time/ gas, etc that I spent to go find it in libraries, courthouses, cemeteries, etc. Had ancestry.com been available when I did this.. I would have welcomed it.
Also at rootsweb... look on the front page for mailing lists, click there and subscribe to one for Italy. Now you have persons knowledgeable about Italy, coming to your email inbox. Not necessarily a website.
Think "records" first, then how you can get the different records. The challenge of the game is to figure out the many different locations that you can find the info. PS.. I have never met a researcher yet who was not on a budget. I used to read documents and study them at 3 in the morning, when I had little ones. It is worth it, by the way.