Should I start with complex variables before fourier series? My math professor has said that it would be useful to start with complex analysis before learning fourier series in the signals and systems course(I'm an undergrad EE). Do you agree with that and why ?! 

(I'm planning to learn complex variables from Brown/Churchil)
Well, Fourier series are more conveniently expressed in complex form as series

$$\sum_{n=-\infty}^\infty c_ne^{inx}\;,\;\;n\in\Bbb N\;,\;\;c_n\in\Bbb C$$

I think complex analysis is way too much for the above, as things can be put in real form in a  more or less easy way, but this may heavily depend on how deep that course of your intents to dig in these matters.

Anyway, a hefty real analysis background, particularly in infinite series, both numerical and of functions, could be enough, together with some basic of complex numbers.
There are two forms of fourier series namely trigonometrical and complex form
As you are an EE undergrad you would be dealing mostly with complex form of fourier series which further leads to development of Fourier transform .

the basic relation that is used is that of euler ie

e^ix = cosx + isinx

and no major part of complex theory is required