So for the relationship between transforms.

All are a member of the set of Orthogonal Transforms, which are orthogonal changes of basis between vectors. It's also a dot product of the original vector and a kernel vector, if you choose to look at it like that.

To be an orthogonal transform, there must exist an inverse transform that maps the transformed vector to the original. In addition, the vectors must obey Parseval's theorem, that is to say energy is conserved by the transformation.

In all fourier transforms, multiplication in the time domain of two functions is equal to their convolution in the frequency domain, and vice versa (with some nuance). In addition, a signal that is discrete in one domain is periodic in the other. Periodic in time = discrete in frequency, periodic in frequency = discrete in time.

The Continuous Time Fourier Transform (often called just the Fourier Transform), maps the continuous time domain to the frequency domain.

The Discrete Time Fourier Transform maps a discrete time domain function to the frequency domain. It is the same as the CTFT, with the caveat that it must be periodic. For convenience's sake, we say that it's periodic with an interval of 2π, which physically maps to the sampling rate.

The hazard with the DTFT is that unless the signal is periodic, it will not be discrete. We cannot store data with infinite resolution. So what we do is sample the DTFT to get the DFT.

The time domain explanation is that we take a finite chunk of the signal and pretend that it's one period of a periodic signal, then use that to calculate the DFT.

Now the nuance with respect to convolution is that multiplication in the frequency domain with the DFT is circular convolution, not linear convolution.

What's interesting is that the DFT gives birth to a subset of orthogonal transforms, called Finite Length Orthogonal Transforms. These are orthogonal transforms with finite length kernels. The special interest of these transforms is that because they obey Parseval's theorem, you can view these transforms as rearranging the energy of the signal. You can design a transform to pack the most energy into the fewest bins, which is useful for lossy compression.

Other properties to be aware of are that these transforms are linear, so some of the properties of linear systems apply (namely associative and commutative properties).