To my understanding, a gauche interaction occurs between two R groups when staggered by 60 degrees in a Newman Projection. When looking down the C-1 to C-2 bond of a cyclohexane, the Newman Projection shows that C-3 and C-6 are staggered by 60 degrees. Would this still be a gauche interaction, and if so, why does it not result in an energy increase in the chair conformation? I was told this is because cyclohexane in the chair conformation acts as a long chain carbon, but I did not understand how that results in no energy increase.
Yes, you're exactly right. Not only is the axial methyl gauche with respect to the methylene unit that you point out, but there are also gauche interactions of the two methylenes that are drawn in the "bridging" positions of your Newman projections.
So, why is the chair conformation so stable?
For one, gauche interactions are really not that destabilizing. You're talking about maybe a few kJ mol-1. Realistically, the axial interactions between that methyl group and other axial hydrogens is going to be more energetically unfavorable. But this is easily solved by a ring flip, which maintains the chair conformation.
Whoever mentioned that chair cyclohexane "acts like a long carbon chain" was probably referring to the fact that the bond angles are close to 109.5 degrees, which means that it is far less strained than rings that are fewer than 5 carbons or greater than 7 carbons. However, that's true for boat cyclohexane as well, so it's not that relevant to the question at hand.
For reference, boat cyclohexane is about 25 kJ mol-1 higher in energy than chair cyclohexane, so the gauche interactions you're talking about are a very minor contribution the the energy of the molecule.
Axial methylcyclohexane has two gauche interactions. One is between the CH3-C1 bond and the C2-C3 bond shown in red. The dihedral (torsional) angle is ~60o. Given the plane of symmetry passing through CH3-C1-C4, the second gauche interaction is on the opposite side of the plane (CH3-C1 bond and the C6-C5 bond). Each of these interactions is worth ~0.9 kcal/mol, a value that is reminiscent of the value for gauche n-butane.
Equatorial cyclohexane has two anti-butane interactions (CH3-C1-C2-C3 (in red) and (CH3-C1-C6-C5). As in the case of anti-n-butane, it is assigned a value of 0 kcal/mol. Thus, the axial conformation is ~1.8 kcal/mol less stable than the equatorial conformation.
Although it is true that there are gauche interactions within the ring, these interactions obviously cancel out in any case because they are the same in both conformations. It is differences in energy we are evaluating, not absolute values. It is the interactions of ring substituents with atoms in the ring that are at issue.