How does one mine for crypto currency?

I know that based on limits of Blockchains for BTC, Ethereum and others currencies, the cost outruns the gains of mining.

I do not intend to make a business of it.

I would like to have "academic" knowledge of it, the purpose would like some reliable current sources on to setup a machine for it.

Which type of tools are used to monitor the contributions made to the network.

As a learning moment, which formulas or problems are we still tryin to solve through shared gpu processing?

Why other systems are not better for those problem-solving?