I studied up on the leaf and shape and color of the figs and now know what kind of tree we have. It's called a kadota fig. The fruit is a yellow/greenish color and typically used for Fig Newtons.
This year I will be very proactive and keep the pests away from the tree. I've gone out this week and picked the eaten fruit off the tree and shooed away as many squirrels as possible.
Here are a few of the ripe ones I picked.
I don't think the crop is completely ready. We have millions of babies just growing, but a few ones are big, soft and droopy. They taste alright, but I can detect a greenish flavor to it. They probably need another month to fully mature. Regardless, I'll be making a ton of fig jelly this Summer.