What are all the ways a group of people can start to pronounce or say things differently? And what are the factors that can affect those "mutations" apart from a random shift in pronunciation of the population that could randomly spread? Surely it's a wide question so I will get by with a wide answer.
I'll go by examples for each question so you get the idea.
What are all the ways a group of people can start to pronounce or say things differently?
While "All" would probably take up a few hundred pages, you could take, say: Palatalization: palatal consonant followed by front vowels do their magic. This happens every day, this is why you pronounce nation the way you do. (https://en.wikipedia.org/wiki/Palatalization_(sound_change))
However, this also happens today when people speak a little faster than usual, or teenagers make things sound cool. So one day you get "gotcha". And it gets picked up by a small group. While they will still know it is underlyingly got you, their children and their children's children will not have that knowledge, for them, gotcha is a word on its own right. And this goes on and on.
Imagine this on the scale of hundreds of phonetic changes taking place within every single tiny community of speakers: family, sports-team, school...
And what are the factors that can affect those "mutations" apart from a random shift in pronunciation of the population that could randomly spread?
Major misunderstanding needs to be corrected here. The changes are not random. They are just about as rule governed as artificial / computer languages. The change is automatic. What is unpredictable is which change will take place among the available set of possible changes for a given phonetic constellation. See Centum vs Satem languages - https://en.wikipedia.org/wiki/Centum_and_satem_languages
What is also unpredictable is which change of the time will be adopted by many enough speakers so its influence is permanent. Fashion, power, rebellion, coolness - all a little fuzzy for linguistics.
Surely it's a wide question so I will get by with a wide answer.
It is. One major force of change, phonetic and language in general, is actual speech. The more frequently words appear in a person's speech, the more they get glued together. Hence, tell hem becomes tell'em. Rock and roll becomes rockenroll, and break fast become breakfast. And there comes a point when they cease to be two independent words, they become one and the powers of phonetic changes begin to work on them.
The other force I had already alluded to is the new generations of language learners. For them a mouse is a computer accessory not a rodent, google is not a number but a transitive regular verb and y'all is the plural of you.