Infinity shares a lot of properties with zero.
0*x = 0
∞*x = ∞
I realize I'm heading into shaky ground with division, but I think this is pretty correct.
x/0 = ∞
x/∞ = 0
Powers:
0^x = 0
∞^x = ∞
Root:
x(sqrt)0 = 0
x(sqrt)∞ = ∞

As for adding and subtracting, let us regard -∞ and +∞ as the same: ∞. Just like -0 = +0.
By adding 1 to 0, we get 1. If we were to add 1 an ∞ amount of times, we would get ∞.
If we were to subtract 1 from ∞ an ∞ amount of times, we get 0.
By subtracting 1 from 0 an ∞ amount of times, we get -∞, or ∞.
By adding 1 to ∞ an ∞ amount of times, we get 0. 0-∞+1, 0-∞+2, up till 0-∞+∞

Some of these need to be rigorously defined a bit more.  For one thing, it is not true that if you subtract 1 an infinite amount of times from infinity that you get zero.  Also, within infinity itself there are cardinalities above Aleph0.  Set Theory is a fascinating field of study, and they define cardinal arithmetic to a tee!

I appreciate your enthusiasm, however!