Where is the fault in my proof? I had some spare time, so I was just doing random equations, then accidentally came up with a proof that showed that i was -1. I know this is wrong, but I can't find where I went wrong. Could someone point out where a mathematical error was made?

$$(-1)^{2.5}=-1\\
(-1)^{5/2}=-1\\
(\sqrt{-1})^5=-1\\
i^5=-1\\
i=-1$$
Your mistake is that you have "$(-1)^{5/2} = -1$". It actually holds that $(-1)^{5/2} = i$ since you get by euler identity that 
$$(-1)^{5/2} = {e^{i\pi}}^{5/2} = e^{5/2 i\pi} = i.$$ 
Furthermore you shouldn't write $\sqrt{-1} = i$ because the root isn't defined for negative values and you can get all sorts of wrong proofs by using the rules for square roots in combination with this notation.
$$(-1)^{2.5} =(-1)^2\times (-1)^{1/2}= \sqrt{(-1)} =  i$$ is your mistake