My attempt: der Anfang meines neuen Lebens.
I just want to make sure I’ve got the grammar perfect here. Also, is der Anfang the best-sounding word here, or do you think der Beginn is a better choice?
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Your grammar is fine, and both "Anfang" and "Beginn" are appropriate. "Beginn" sounds a little better to my ears, especially in connection with "neues Leben" (new life). I think it's because "Anfang" is a more formal word than "Beginn", and "Beginn" is more often used in a poetic context. Google seems to agree, with "Beginn meines neuen Lebens" having three times more results compared to "Anfang meines neuen Lebens".
I'd like to add yet another aspect:
"Beginn" is more formal and seems to refer to a point in time, where something starts.
"Anfang" is less formal and seems to refer to the initial period of something.
Note: This may be super subjective - I suppose the points this answer gets will tell if anybody else feels like this as well.
der Anfang meines neuen Lebens
seems okay to me. I agree with the other answers. Yet I think
Anfang can be more substantial. It can also imply something like origin, whereas
Am/Im Anfang schuf Gott [...] (Gen. 1, 1)
Ich bin [...] der Anfang und das Ende (Off. 22, 13)
Rauchen macht sehr schnell abhängig: Fangen Sie gar nicht erst an!
There is more in
anfangen than "smth new starts" while it indicates itself something new is coming up. It can be used more widely, while
beginnen might sound a bit uncommon sometimes and slightly indicates (or is often used), when one talks about something which obviously will have an end
Der Unterricht beginnt um acht und endet um eins.
That's how it might say in a written guide about the lessons. Colloquial that'd be
Der Unterricht fängt um acht an und geht bis eins.