I'm having trouble understanding infinity. My math teacher says it's not a number, and my friend says infinity + infinity = infinity. I'm just really confused at the moment. I don't get why this is, would you be able to explain it to me please?
Sure, i will explain it in general and then more precisely.

Infinity is a concept for something endless.
It roughly means, that you can't directly count or measure something. 
For example, you can count the natural numbers between 1 and 3 but you can't count all natural numbers.

There are also different kinds of numbers.
Natural numbers: {1,2,3...}
Whole numbers: {..., -1, 0, 1, 2,...}
Rational numbers: {-1/9, 3/9, 1, -2, 1/3, 0.333, ...}
Real numbers: {pi, squareroot of 2, -1, 1/3, ...}
and more

It is better to start by learning the simpler kinds of numbers.
So when your Teacher says that infinity is not a number, its the simplification that infinity does not fit within these kinds of numbers.
When you get deep into studying math, you can define infinity as a Hyperreal number, but this is quite advanced.

Regarding to your friends statement:
Because infinity is not a real number, you can't just use the same real number addition operator (+) like: infinity + infinity = infinity.
You can use the mathematic definitions of a theory called cardinal arithmetic to generalize the addition operator (+).
So infinity + infinity = infinity is true, if you clearly state that you are using cardinal arithmetic.
But you have to be remind yourself, that the addition relies on additional definitions and is not quite the same addition as in 1+2.2=3.2.

In fact, there are even multiple kinds of infinity, but most times we talk about the infinity called aleph-0.

Infinity is such a useful concept, that its even used to define the core assumptions or axioms of Math.
For example the Axiom of infinity in Zermelo-Fraenkel set theory, the most popular fundamental Theory of math.
can you explain Zermelo-Fraenkel set theory more thoroughly