In my notes the lecturer takes the Fourier transform in $x, y$ and $t$ of $\phi(x,y,z,t)$ as: $$ \int_{-\infty}^{\infty}dt\, e^{i\omega t}\int_{\Bbb{R}^2}\,dx\,dy\,e^{-i(k_1x+k_2y)}\nabla^2\phi(x,y,z,t)=\left(\frac{\partial^2}{\partial z^2}-k^2\right)\widetilde{\phi}(k_1,k_2,z,\omega) $$ but how does he compute the integral? I would've guessed integration by parts but how do you integrate, for instance, the $\partial_{xx}e^{-i(k_1x+k_2y)}$ term?
On a related note, how do you take the gradient of a function that depends on $x, y, z, t$. Do we just get $\nabla=(\partial_x, \partial_y, \partial_z,\partial_t)$?
Yes, integration by parts is a straightforward way to go about it (assuming the function is integrable, the boundary terms vanish, etc. which is often the case in these sorts of problems). I would suggest starting with the simple case $$\int \mathrm{d}x\,e^{-ikx}\partial_x f(x)$$ to see how it works. You will get a factor of $\partial_x e^{-ikx}$, which you can evaluate directly - it's just a derivative - and the resulting integral can be expressed in terms of $\tilde f(k)$.
This shows you the general principle that a Fourier transform turns differential operators into factors of $-ik$. Once you've verified $\partial_x \to -ik$ from the simple example, you can hopefully figure out the derivation for $\partial_{xx}\to -k^2$. Also, it shouldn't be too hard to generalize this to multiple dimensions. Remember that $$\nabla^2 = \partial_{xx} + \partial_{yy} + \partial_{zz}$$