So I know that in order to prove it is a group, there are several things that have to be confirmed, including associativity, closure, inverses, and identity. Associativity seems like it can be assumed, but the others till have to be proved. I'm having most of my trouble with the closure and inverse proofs, especially that for closure.

Why did you delete this? I wouldn't say you should
assume associativity but certainly you can just note that multiplication of real numbers is associative and this is just a subset of the real numbers. To show closure write the product of two such numbers as [itex](a+ b\sqrt{2})(c+ d\sqrt{2})[/itex] and actually do the multiplication. What do you get? Show that it can be written as [itex]u+ v\sqrt{2}[/itex] by showing what u and v must be. The identity is [itex]1= 1+ 0\sqrt{2}[/itex], of course.
And the multiplicative inverse of [itex]a+ b\sqrt{2}[/itex] is [itex]1/(a+ b\sqrt{2})[/itex]. Rationalize the denominator to show how that can be written in the form [itex]u+ v\sqrt{2}[/itex].