What & where to go for 15 days in Europe free time

Consider Romania, we did a six day bus tour there after out Danube cruise last year and it was great and not expensive.

Take a look at Gate 1 Travel, they have great tours with low prices. Perhaps they have a tour that would include Croatia and/or Slovenia.

How long are you planning to stay in Budapest following your arrival? It’s worth at least 4 days of your time before heading off.

The rest of any suggestions I might make would depend on time of year.

Look and see which locations share Wizz and Ryan service from Rome.

For instance Wizz and Ryan both fly Budapest to Malta, and Ryan flies Malta to Rome.

Better for my taste, Budapest to Podgorica on Wizz (thursday and sunday), then Dubrovnik to Rome on Ryan.

Or instead of flights, you could look at what cities you can get to by direct train. Here is the info from Budapest:
https://www.seat61.com/international-trains/trains-from-Budapest.htm

I find it a little odd to call out "mosques/temples" as something you don't want to see, particularly when there aren't a huge number of these in Europe compared to other regions.

Better for my taste, Budapest to Podgorica on Wizz (thursday and
sunday), then Dubrovnik to Rome on Ryan.

You didn't give dates did mention summer and this works best May through September. im with you. Avoid Croatia in thr high season.

The implied suggestion since you mentioned Croatia but wanted better costs and fewer tourists was to spend the time in Montenegro, which is sort of like Croatia before the tourists swarmed it. You end in Dubrovnik that despite the tourism and price is every bit worth it and has the flight to Rome.

Between Podgorica and Dubrovnik you travel the coast from Ulcinj to Stari Bar to Budva. Maybe run up to the Tara River Caynon to see thr deepest canyon in Europe and cooler weather, zip line across it, raft thr river. Then down to the Ostrog Monestery built into he side of a cliff followed by views fom the best restaurant deck in the Balkans. Finally, over the mountain and down to the city of Kotor, then along the bay to Perast, Herceg Novi and on to Dubrovnik.

There is another post right now where I believe the OP is contemplating 12 days across Croatia in a rental car, which I believe will cost him $1000 to $1500. You could hire a private driver in Montenegro for what i just described for less ... I'm pretty sure.

I will be back doing something similar, Montenegro trip 5, in the spring.

I use travel videos to help decide where I want to go. This one has links to Montenegro videos. https://community.ricksteves.com/travel-forum/montenegro/podgorica-to-dubrovnik-by-car

I always thought comments on Croatia being expensive were because people just assumed it would be cheaper. I found there was great value for our money there. Switzerland was by far more expensive.
Why not Slovenia and Italy?

You are correct, Croatia is indeed less expensive than the most expensive country in Europe. Here is a little on the problem: https://www.croatiaweek.com/too-expensive-for-tourists-why-croatias-prices-are-raising-concerns/

But it did do a good thing. It cracked the Balkan tourism shell. Now we have BiH, Montenegro, N. Macedonia, Albania to explore.

Croatia is by far not the most expensive country in Europe. Especially compared to Switzerland and Scandinavia.
I will admit that Dubrovnik prices have tripled in the past few years, especially after they went to the Euro. But the rest of Croatia is very affordable.

One of the problems is that it became too popular due to Game of Thrones sites and social media. The overcrowding is due to all the cruise ships.
One good thing about Croatia’s coastal cities for travelers, such as Dubrovnik and Split, is that they are what I categorize as a one and done city. Once you have visited and seen the city, which is a living museum, there is no need to visit again.
The one bad thing is that they are a PIA to get to.
Where ever you decide to go, have a great trip.

While Croatia prices have increased, I think you'll find it still cheaper than other parts of western Europe if you do it right. You'll probably find it much cheaper than either Italy or Switzerland, that's for sure. A good friend of mine highly recommends Croatia and I think June is a great time to be there, but will be busier. You could go to Venice from there (highly recommend) and spend a few days and train through Italy.

Barbara, I agree with you on the one and done. Still for some reason I have been back to Dubrovnik a few times and I havent hated doing it. Split? Twice and you can have it. Didnt enjoy it at all. But one of the attractions was the cost. Something new for a bargian. "New" was important because as you point out these are sights you will want to return to. Pleasant, enjoyable, somewhat interesting, but not drooling wow!!! But for the price, it was very okay. Not sure thats still true.

Saying Croatia is cheaper than Switzerland, the most expensive destination in Europe, isnt really saying anything to build confidence that Croatia is reasonably priced. Maybe it has only come up to match Italy, but Italy is a destination to drool over and that makes the cost a good investment.

The articles say that tourism costs over the last 3 years in Europe on average have grown 15 to 20% while in Croatia they have gone up 40 to 50%. I was Croatia about a month ago and I was stunned at the costs. Substantially more than Budapest. But I had a good time. No regrets. I rememeber my first trip to Croatia about 10 years ago, it was indeed a bargian. As cheap as Budpaest. Heck it was still cheap in 2021 when I was there. As for the crowds. Wow! Zadar was packed. Zadar!! Who goes to Zadar. Apparently everyone now. I went to an obscure island for a few days and missed most of the tourists that way.

Maybe the new Croatia is Montenegro down the coast and Albania and maybe BiH. But in some number of years maybe thats not worth the $$$ either.