Go to the very easy-to-use Deutsche Bahn website for schedules/fares/tickets between Berlin Hbf and Braunschweig Hbf. The various departures take between 1-1/2 and 2 hours. I simply choose on the basis of duration, time of departure and fare. You can potentially save quite a bit by buying this ticket soon if your departure date and time are rock-solid; the cheapest tickets cannot be canceled. It looks as if today's price for many May 24 trains is 30 euros. Buying at the last minute could cost your around 80 euros.
The trip from Munich to Naples will take 11 hours or longer. You can see schedules but not fares on the Deutsche Bahn website. I'd strongly suggest flying. You can check skyscanner.com for flight schedules and fares. Be aware that there will be fees to check bags and there may be tight weight and size limits for cabin baggage if you fly a budget airline.
For the Italian trains you have a choice of Trenitalia or Italo. Similar speeds; I'd choose the time/fare I liked. I'm not sure about the Italo website, but on trenitalia.com you'll need to use the Italian station names. The fast trains usually go only to the main train station, so those are the ones I've listed below:
Munich = Monaco di Baviera
Naples = Napoli Centrale
Rome = Roma Termini
I can see the Munich-Naples connections on trenitalia.com. Again, buying early will get you a nice discount if your travel schedule is solid.
There are some good deals on non-refundable tickets available for the Naples-Rome run as well.
You will definitely not benefit from a rail pass if you're prepared to buy the tickets soon.
Edited to add: When you buy a ticket on one of the fast Italian trains, it will come with a seat reservation included. In Germany, many of the trains do not require seat reservations, but you can obtain them on the fast trains--I think for a modest extra charge. I would certainly want to have a seat reservation for the German train that makes up part of the Munich-Naples trip.