Suppose I have a setup where there are 2 mirrors parallel to each other and I let in a laser beam of a single wavelength of red color through a tiny little hole in one of the mirror so that the beam would strike the other mirror at right angle. Suppose the laser bounces back and forth and the mirrors remain perfectly still, how do I check if there is destructive interference or not?
If you send a beam of atoms perpendicular to the laser bouncing back and forth such that these atoms have some absorption line in the frequency of the laser, then if the interference was destructive you would see the atoms coming out exactly as they have came in (because ther is no light to absorb). Otherwise, if the interference was not destructive at all, some atoms will end up in some excited level that you could measure.
I will keep my answer general in hopes that it will be relevant to your question. You should recognize that if you reflect a beam off of a mirror, you will get a standing wave, which is the result of the incoming wave interfering with the outgoing wave. A standing wave has "nodes", which are points of destructive interference. For instance, for a perfect metal mirror, the electric field at the surface is zero at all times due to perfect screening currents. So you have destructive interference there, as well as at every half wavelength away from the surface (for normal incidence). There is always destructive interference.
But you also asked about what seems like a Fabry-Perot cavity. In this situation, you get optimal destructive interference (as well as constructive interference, at different positions) at the resonance condition. How can you tell that you are at the resonance condition? There are many ways, but the most obvious is to look at the light transmission through the (not quite perfect) mirror. If you get a peak as a function of mirror position, you are at resonance.
A schematic diagram of an arrangement which will enable you to view the interference fringes might look like this.
Please note that to see the fringes as described below the mirrors have to be optically flat, in other words flat to within fractions of the wavelength of the light emitted by the laser.
You will note that the inclined mirror reflects light from the laser down onto the parallel mirror and then allows light which is reflected from those parallel mirrors to pass through and hit the screen where the interference fringes can be viewed.
In the arrangement as shown you will see circular fringes whose radius will vary as the separation between the mirrors varies.
As the separation of the mirrors is decreased fringes seem to be "created" at the centre and appear to move outwards from the centre.
When the parallel mirrors are very close (a few wavelengths) to one another then the radius of the fringes becomes larger but you are unlikely to see a uniform illumination (or lack of it) across the whole screen.
The width/sharpness of the fringes will depend on the number of significant reflection there are between the mirrors and as the reflectivity $R$ of the mirrors goes up the fringes become sharper.
This increase in sharpness as there are more significant reflections between the parallel mirrors is equivalent to the increase in sharpness in the fringes produced from multiple slit interference as one goes from two slits to many thousands of slits which is called a diffraction grating.
A simple experimental set up to observe such fringes using two microscope slides is described here with the separation of the two reflecting surface (the sides of a microscope slide) changed by heating the microscope slide with a hair dryer.
The reflectivity of microscope slides is low so to achieve the very sharp slides you need to use a Fabry-Perot etalon which is described here.