I will be flying Turkish Airlines for the first time in January and it will also be my first time at such a huge airport. I have about 1h 40m layover time which I hope will be enough for me to get off my first flight and get to my gate, but what would happen if, for example, I miss my boarding group? Can I board with the one getting on the plane at that moment, or wait for everyone else to board and then be the last one? Does anyone have any experiences with this?
There's no problem. If they call to board group 3, for example, anybody in groups 1, 2, or 3 can board (but not people in group 4 yet). So if you miss your boarding group, you can just join the line and board with any later group with no problem.
The worst-case scenario isn't bad at all: if you get in line too early (say you're in group 4 and they're only boarding group 1), you may be asked to step out of line and wait when you reach the front. So even if you make a mistake, nothing bad will really happen.
Note that you must be at the gate 20-25 minutes before departure, so don't plan on showing up at the very last minute.