NYE Europe Itinerary help

Any advice?

Thermal underwear and layers.

I always enjoy second-guessing assertions that a multi-destination air ticket is more expensive than a simple round trip.
Not knowing your departure airport, I used Atlanta because it is a major gateway. On the dates you show, a multi-destination itinerary from there into Prague and back from Budapest will cost about $150 more than to Prague and back round-trip.
That saving will be eaten up by the cost of returning to Prague, losing most of a day on an already tight time schedule. There could be a wide variance depending on your point of departure, of course. You may already have researched that factor.
My standard advice is to be sure to use a multi-destination search function as these itineraries are not a pair of one-way tickets.
(PS: I used matrix.itasoftware.com for information.)

Skip Munich. Share a guide book with your crew to see what strikes your interests as far as site seeing goes. I have traveled with people and priorities differ and occasionally we split up or go out separately. Also look up the city tourist information sites for each city; all the cities you mention have one all you have to do is Google tourist information and the name the city, that way you know what is going on cultural events when you get there.

I work in academia and have gone to Europe every year for the last five over Xmas break; it is one of my favorite times. The small tourist attraction towns (daytrips) is what you need to research closer as stuff closes for the season. You can look up hours on web for every tourist site.

Ha! Southam, I appreciate the second-guessing! At the time I was looking I did try the multi-city option and could find nothing favorable (considering our desired departure times, due to family Christmas obligations right before the trip) under 1300 (I even tried the double one-ways, the roundtrips + intercity flight). Now I can see slightly more favorable options around 1100-1200 for multi city, though the departure times on both ends are not optimal. Perhaps I should have waited a bit more, though I thought the holiday prices would not go down. We are flying out of Los Angeles so the Prague roundtrip we found was 850 through Air France, though it appears to have disappeared now. I'll concede that if we were willing to do some very early mornings to catch flights, the cost difference for the multi city might be worth it considering the day saved and the extra train fare. But a good thing to keep in mind for next time! Must remember to use matrix.ita, too, though I typically use google flights and they use the software, I believe?

Jazz+Travels: Thanks for the advice! Just curious, why do you say to skip Munich? I think as a city itself, it is not as appealing, but the thought was to head in that direction to be able to maybe do a quick stop in Bad Ischi or Hallstatt, and see neuschwanstein castle or maybe a glimpse of the alps. More as a base for some daytrips. We will definitely be doing research on the cities themselves, but could see benefits to either itinerary. I always scour old forum questions and answers (fully aware that everyone has their own preference) and have found anecdotal information really helpful, that's why I thought I should ask here.

Also, joe32F: Noted. We are weaklings from southern California so there will be layers aplenty.