[CheezeSlayer]

I'm having a hard time with this particular chemical equation that I am supposed to balance.

Cu(s)   + H + (aq) + NO3-(aq)   --> NO2(g)   + H2O(l) + NO3-(aq) + Cu2+(aq)

The instructions are the balance the Net Ionic Equation of this chemical equation. 

I know that in a properly balanced equation that no same species should appear on both sides of the reaction so I cancelled out NO3- on both sides and was left with:

Cu(s)   + H + (aq) --> NO2(g)   + H2O(l) + Cu2+(aq)

After balancing this equation my answer was:

Cu(s)   + 2H + (aq) --> NO2(g)   + H2O(l) + Cu2+(aq)

I'm not sure if this is correct, can someone please provide guidance?

[UG]

Have you heard/learned about balancing redox half equations? You can balance this reaction using this method.

[Borek]

I know that in a properly balanced equation that no same species should appear on both sides of the reaction so I cancelled out NO3- on both sides

That's not correct. That is, you are right that no species should end on both sides, but NO₃^- gets reduced to NO₂, so you can't remove before balancing.

Cu(s)   + 2H + (aq) --> NO2(g)   + H2O(l) + Cu2+(aq)

I'm not sure if this is correct, can someone please provide guidance?

Simple check - balanced reaction has the same number of atoms and the same total charge on both sides. Do you have the same number of atoms of oxygen and nitrogen on both sides?