So I'll try my luck here too:
anfangen and beginnen are definitely the same in meaning but beginnen generally sounds a bit more "formal".
I think anfangen is the way to go in every day situations.
It starts raining.
Es beginnt zu regnen.
Es fängt an zu regnen.
I think the second German phrasing is more common as the event is so mundane that it doesn't really merit a beginnen. A play in a theater beginnt.
Angehen has to be treated with care as there are 2 versions in sense of to start.
Der Motor geht nicht an.
The engine won't start.
This is NOT connected to a personal effort. But it is however limited to a machine sort of thing. It doesn't work for rain or a play so it is clearly different to the first 2 words.
The second angehen is
Ich gehe ein Problem an.
That does not mean that I start it but that I start dealing with it. So it has some notion of to start but it is definitely way different to beginnen, starten, anfangen and the other angehen.
Just for completion there is also losgehen, which is to start but by oneself. So a movie can losgehen but I can't losgehen to read.
And then there is anmachen, which is a bit like starten but even more limited to a machine.
Ich mache den Motor an.
I start the engine.
Now let's get to stop. The verbs suggested are NOT THE SAME as landei supposes... especially not stoppen and aufhören.
Stoppen means to stop a motion or process of some kind and it is a rather sudden and definite stop in comparison to the others.
Stoppen is usually not done by the thing itself but by someone. You can use it that way but it is rare. Generally it is not a word I use very often.
If I have to stop someone from doing something I would prefer aufhalten but stoppen works too.
Anhalten is the best choice for locomotion. That is pretty much it but that makes anhalten a very frequently used term already.
aufhören can be used for all kinds of actions BUT only you yourself can aufhören doing something. You cannot aufhören a car. That will not make any sense. I have written an article on aufhören on my blog and one about einstellen if you want to know more about that word. Aufhören has no notion of finishing. You just stop and might start 3 minutes later again.
beenden includes finishing. It is always you who beendet something. Things cannot beenden by themselfes. The proper word for that is enden.
Der Film endet um 10.
Ich beende meinen Vortrag.
I might be back for edits but I hope this was somehow clarifying things.