For filters with impulse responses on a specific $x[n]$, how do you calculate DFT of $y[n]$?
How would you find Yk (the coefficients of the DFT of y(n))?


By applying the DFT to Y.

So: math.

Mathematically, you filter something with a FIR by convolving the signal with the taps vector:

$$y = x * h$$

Hence, the DFT of $y$ becomes is:

$$\mathcal F \{y\} = \mathcal F \left\{ x * h \right\}$$

and thanks to the properties of the DFT, convolution in time domain becomes multiplication in frequency domain:

$$\mathcal F \{y\} = \mathcal F \{ x\}  \mathcal F \{ h \}$$

So you just apply the DFT to both your signal and your taps vector and multiply.

Seriously, this is very basic DFT theory, and you will learn that if you continue your studies of the discrete Fourier transform.