I would use the first few pages to draw a family tree, starting with yourself on the left side of the pages, and branching out to parents, grandparents, etc. As you confirm people, you can fill their names in on the main chart, used for quick reference.
I would organize using family group sheets. Each of these contains:
Husband: Born, Married, Died, Father, Mother
Wife: Born, Died, Father, Mother
Children: Name, Born, Died, Married, Spouse
Assign a number to each family group sheet, so you can make easy references to other sheets (ie. everyone will appear on at least 2 family group sheets).
Use tabs to divide the notebook into 5 sections: Main family tree, you, your parents & your grandparents family group sheets at the front, then a section for each of your grandparents lineage.
Also, here are some links to blank forms you can print out and use for doing your research:
http://www.cs.williams.edu/~bailey/genealogy/
http://www.familytreemagazine.com/forms/download.html
http://www.free-genealogy-forms.com/content/blogcategory/6/32/
http://www.ancestry.com/charts/ancchart.aspx
***I would strongly suggest just taking notes (I use some of the forms above, and just keep them in the order I'm working on them, in folders, labelled by family surname) and then entering the data into family tree software on your computer on a regular basis. MyHeritage has free software that is really good, and can also help search the web. You're going to end up with A LOT of information. As an example, my husband's maternal grandmother's lineage is 90 pages, printed off the computer (and that's just the basics, their is much more information on the computer!). His paternal grandfather's is about the same. So, you have four grandparents lines to do x 90 pages...that's 360 pages! If you're married and do both yours and your spouses', that's 720 pages! Just a basic idea of the amount of info you're going to have to keep organized.***
Ultimately, you'll need to find what works best for you, but hopefully I have provided some useful tools & tips to get you started.
Good luck in your research, however you choose to organize yourself!