Setting up a simple "manual" neural network: Feasible?

I want to read up and familiarise myself with the developing world of artificial intelligence - not with the aim of landing a job in that field, simply out of a personal layman’s interest and to have a project to while away lockdown time meaningfully. As part of that, I’m thinking of setting up a simple neural network and train it to perform some simple task, like assigning data points to categories. It doesn’t have to be anything productive or useful, but it should be something that makes you go “Wow, I wouldn’t have thought a machine could learn to do that”.

I know, of course, that there are plenty of software packages that can do this, but in order to get a feeling for the mechanics going on in a neural network, I’d like to do it largely manually. By that I mean that I either run the entire network manually (drawing up a diagram with the neurons and the links between them, and manually calculating the inputs, weights and outputs for each neuron), or to code it in Excel in a way where the calculations going on in each neuron, and the way one neuron affects the next, is plainly visible. Kind of like these paper simulations of very simple computers that you sometimes see.

Am I fantasising here? Is that a realistic goal? My fear is that I’d be caught in some kind of trade-off where anything that a neural network can do is either so complex that it couldn’t be done in my manual or semi-manual environment, or so trivial that it can be done but be entirely unimpressive. Obviously I would also need to train my network, for which I would have to find a way to implement a backpropagation algorith in this manual or semi-manual way.

I suppose I’m not the only novice to AI who’s thinking about doing something along these lines, so I’d appreciate people’s views as to whether this is feasible. If it is, I’d also be grateful for resources such as websites or books that could help doing it.

Here’s an introduction to neural network that involves programming single “neurons” in Python. Does that get to what you’re after? I stopped reading early, so I don’t know if it actually gets to the point where it solves any interesting problems.