On your computer it is really easy to create a new folder for each month. I recommend naming them "201X-XX". Put the year first followed by the two digit month. For example October 2012 would be 2012-10. This will make all your months show up in order as opposed to alphabetically where August will always come before February. This also keeps entire years together.
This system works best for those people taking basic everyday family photos. If you need a outside summertime photo for a scrapbook page simply look in June, July or August's folder.
Flash drives or thumb drives work great for people who take a lot of photos at one time. For example if you are taking photos of a special event such as a wedding just load all those photos onto one flash drive.
Another way to use these devices is to put all the photos for a month, quarter or year on one flash drive - depending on how many photos you take and how much the device will hold.
If you are the type of person to get every photo developed than a photo box is the system for you. Separate the photos by month, quarter, year or even by subject or project. An example would be all Christmas photos in one box or all photos of little Susie in another box.
On a side note if you use a digital camera I highly recommend only printing out the photos you will actually use. Making sure you will be using them in a scrapbook, picture frame or other craft project in the near future. Wasting less ink and paper.
There are three really great sites online that will host all your photos for you. There are lots of sites out there, but these are the ones I've tried and would recommend.
This is easy to use. There is a 300MB bandwidth limit each month with the fre*e account. You can upgrade to PRO for $1.87 a month. It's pretty inexpensive. My only issue is you need to set each photo to copyrighted otherwise it can be used by anyone under the Creative Commons License. I also recommend setting each photo's privacy settings so it can only be seen by certain people such as Friends or Family.
This is another easy to use site. There is a unlimited photo uploads and 500 video limit for the free account. There appears to be a 10GB monthly bandwidth limit. The copyright settings appear to be about the same as Flickr.com's. One feature I really enjoy is you can mark all photos so they cannot be copied or shared - you don't have to mark each individual one.
I love this site. Perfect for your 365 Photo Project. Simply upload a photo onto each day on the calendar. They even give you the option to order a printed daily calendar (month at a time or whole year). Lots of fun!
The main thing is to find one organization system that works for you and your photo storage needs.