I just posted on this topic on my blog: History on Kindle: Today's Highlight: Genealogy
The books are ones on Kindle but they are of course mostly available in paper format as well, aside from some of the public domain ones mentioned.
The first place to start is with the oldest members of your family, hopefully a grandparent but if not, a parent. Get their vital data/BMD (Birth, Marriage, Death), as well as that of their parents and if possible, their grandparents (and their spouse, if necessary). That should give you something to start with. Before investing a lot of money with a site like Ancestry.com - check out www.FamilySearch.org
, they have the largest free records database on the internet. If you're still interested, check out Ancestry.com's free 14 day trial - FamilySearch has a great free database but it's very limited compared to Ancestry's. That is the easiest and cheapest way to get started. You may find you want to pay for Ancestry in order to continue your search (not everyone can find everything in 14 days and don't forget that they are constantly added more records all the times). The books I posted in my blog should help you get started too.
Keep in mind a few things:
1. Other people's trees can be and quite often are wrong! Do not take information from them unless they have accurate records to back it up (and if you do take info from their tree, a "thank you" is very much appreciated, the genealogy community can be a little precious about "their" info and many privatize their tree so people can't "steal" their work).
2. Records can be wrong - especially censuses. Names are frequently spelled wrong and ages/dates can be slightly off, birth locations can be just plain wrong. And I don't just mean the transcriptions - the original documents can have errors too.
3. Ancestry.com's "hints" can be wrong so always verify the record before added it to your tree.
4. Because of all this, don't assume something is true until you have 3 separate sources to confirm it (and 3 member trees don't count, they were probably all copied from each other).