Given the balanced net ionic equation

Zn(s) + 2H^+(aq) <---> Zn^2+(aq) + H2(g) would the cell notation be:
Zn|Zn^2+||2H^+|H2

(i'm not sure if hydrogen gas should be in the cell notation)

What you have is ok under most circumstances. The molarity of the Zn ion and H ion usually are shown. Usually we write (g) after H2. Finally, usually we add a comma and Pt since H2 gas is non-conductive. If these were 1 M concentrations I would do it this way.

Zn|Zn^2+(1M)||H^+(1M)|H2(g),Pt

To determine the cell notation, we need to understand the components involved in the balanced net ionic equation.

In the given equation, Zn(s) represents solid zinc, 2H+(aq) represents two aqueous hydrogen ions, Zn^2+(aq) represents an aqueous zinc ion, and H2(g) represents gaseous hydrogen.

The cell notation is written in the form:
Anode | Anode solution || Cathode solution | Cathode

In this case, since the zinc (Zn) is being oxidized and is the source of electrons (anode), it will be placed on the left side of the cell notation. The zinc ion (Zn^2+) is in the cathode solution and is being reduced, so it goes on the right side of the cell notation.

Thus, the correct cell notation would be:
Zn(s) | Zn^2+(aq) || 2H+(aq) | H2(g)

The hydrogen gas is included in the cell notation because it is one of the species involved in the reaction at the cathode.