I'm trying to understand 2d Fourier Transforms as applied to images.
Given an image, is a 2d transform equivalent to taking an infinite amount of slices of the 2d image to form a one dimensional function and then applying a 1d Fourier Transform on each slice/one variable function?
Also, if you can, could you also explain how my description would be represented algebraically, and if there are any relations to the 2d Fourier Transform (if my explanation is inaccurate)?
No; it's a composition of $2$ different $1$-dimensional Fourier transforms, viz. $$\int dx dy f(x,\,y)\exp i(kx+qy)=\int dy \bigg[\int dx f(x,\,y) \exp ikx\bigg]\exp iqy.$$