We took everything out of the gift bags and put it on the floor in piles where it stayed for a week or so until I could bring myself around to putting it away. I couldn't really come up with a good solution for the organization so my solution was avoidance.
Eventually I figured out the best way to organize everything and I can proudly show you all what it looks like now!
First thing I did was purchase under bed storage for anything that she won't be able to use in the first three months. I have three bins and I separated everything in groups: 3-6 months, 6-9 months, and anything over 9 months. In these bins, I put mostly clothing but I also put in toys and cloth diapers. The bins I bought look like this.