For Fourier methods, you always need the characteristic function of the log-asset price $\ln(S_t)$. In the Black-Scholes model, $\ln(S_t)\sim N\left(\ln(S_0)+\left(r-\delta-\frac{1}{2}\sigma^2\right)t,\sigma^2t\right)$. It is well-known that the characteristic function of $X\sim N(m,s^2)$ is given by $$\phi_X(u)=\exp\left(imu-\frac{1}{2}s^2u^2\right).$$ You can derive this by a simple integration exercise. As you said, it's the Fourier transform of the Gaussian bell curve. This function is, of course, complex valued.
As @LocalVolatility pointed out, you may need the characteristic function of $\ln\left(\frac{S_T}{K}\right)=\ln(S_T)-\ln(K)$. In general, for any constant $c$ and integrable random variable $X$, we have $$\phi_{X+c}(u)=e^{iuc}\phi_X(u).$$
Fang and Oosterlee derive $V_k$ for some European-options and demonstrate a way of estimating $a,b$ based on the cumulants of the distribution. Having found all of this, the implementation is very easy. According to Hirsa (2013), the COS method is the ``fastest known Fourier-based method''!
Carr and Wu (2004) and Lewis (2001) list characteristic functions for many different exponential Lévy processes (e.g. Merton, Kou, NIG, VG, CGMY, ...). Stochastic volatility models such as Heston (recall ``little Heston trap''!), double Heston, 4/2 have closed-form characteristic functions as well. Even characteristic functions of rough volatility models can be approximated. Some models however do not have a known characteristic function (e.g. CEV, local volatility). So, you cannot use the COS method for these models.