so e^(σ + iω) = e^σ * e^iω is just an exponential growth or decay modulated by a sinusoid.. or, if σ is one, is just a pure oscillation

ω is the usual frequency, but σ + iω is the complex frequency. the fourier transform deals with function that receives ω as input, and the laplace transform deals with functions that receives σ + iω instead.

so the fourier transform is just a special case of the laplace transform with σ = 0

Another useful way of looking at it: Laplace transform is doing many extra Fourier-transform-but-with-decay giving you a map of which "global decay timescale" fits your data best -- since each "slice" is itself sufficient to fully describe the time series

They are all cases of integral transforms with different choices of the set of "primitive fingerprints" -- see chirp transform, wavelet transform, chirplet transform etc -- all taking advantage of the fact that if you choose one set of basis "brushes" that are not redundant with each other (e.g. having red-green-blue brushes is independent, as is magenta-green-yellow but having red-green-blue-yellow is not) then you will be able to describe your signal in terms of a composition of those kernels.

> a map of which "global decay timescale" fits your data best

What do you mean by this?

> -- since each "slice" is itself sufficient to fully describe the time series

But each slice is multiplied by an exponent.. which, okay, becomes a convolution that lets you recover the original function