Symmetries of the v=0 and v=1 quantum harmonic oscillator wave functions and those of the Cartesian Coordinates (References)
In general you need to understand point group tables. 

These always have the same form, which is to say that symmetry operations are on the top line, characters (1,-1, 0 etc) in the main body. The most left hand column the Mulliken symmetry labels (also called symmetry species) for each irreproducible representation (one row of characters) . 

The right hand two columns, often without header labels contain the operators for different type of transitions. x,y, z are used to dipole transitions because a dipole transforms as x, y or z. (The z direction is usually that of the principal axis). Rx, Ry, Rz are used, for example, spin orbit operators (These are coupling between electron spin and orbital motion, Not rotations). 

The last column contains product operators $x^2$, $x^2-y^2$ etc and are generally used to describe Raman transitions since the Raman transition operator depends on a molecules polarisability (a volume) which in projection is proportional to an area hence squared terms. 

If you want more complicated operators say $x(x^2-y^3)$ then generally these can be made from simpler operations. When making more complex functions consider the x as a $p_x$ orbital with two lobes of different 'colour'. The subtraction as in $x^2-y^2$ then looks like the $d_{x^2-y^2}$ orbital with opposite lobes coloured in the same way but each pair of a different colour. You can the use the symmetry of this object to determine which irreproducible representation it belongs to. In $C_{2v}$ (below) it is $A_1$.

The C$_{2v}$ point group shows these features. The operators are E, $C_2$ etc,. The Mulliken symbols for the symmetry species, in your case vibrational normal mode symmetries, are $A_1$, $A_2$. The characters are 1, -1 etc.

A totally symmetric normal mode $A_1$ has characters 1,1,1,1, is active to transitions with a dipole on the z direction and is also Raman active, as it has at least one squared term, x$^2$ for example. The $A_2$ symmetry species in not ir active, but is Raman active, due to the xy operator. You can follow the rest.



If you want to work out what the symmetries of a wavefunction are in a given point group, then the easiest way is to sketch the wavefunction and apply the symmetry operations from that point group. Choose a set of axes, z vertical and x horizontal and y out of the page, perpendicular to the other two. Count 1 if the symmetry operation is present and -1 if not. 

For example the $v$=0 wavefunction for the harmonic oscillator has a gaussian or bell shape. By applying in turn the symmetry operations in C$_{2v}$ this wavefunction belongs to the $A_1$ symmetry species. The v=1 wavefunction has a node at the origin with an inversion in the values between negative and positive x. In this case C$_2$ is -1, $\sigma _{xy} = 1$ and $\sigma _{yz} = -1$ which is the $B_1$ symmetry species.