Family of three looking for the right city

Hello! My husband and I are in our 20s and we have an 11 month old. We are looking to move to Europe, a point of interest is Athens, however this website didn't give me the option to make a general post I had to pick a specific location so I'm sorry this isn't all about Athens, I hope you can still help us! Obviously our plans depend on the status of the current pandemic. We plan to spend 90 days in Europe in less than a year so we can travel around and explore our options. What we're looking for:
Somewhere we can live (family of three) on $3000 usd a month (or less)
We'd want to be in the city center, a two bedroom., furnished.
We don't want a car, we'd need to rely on public transportation
We'd prefer mild weather, open to cold weather but I'm just afraid of intense winters. Close to skiiing, easy travel within the city.
We really enjoy the outdoors, we want to walk everywhere, I'd love to hike everyday if possible!
It would be a huge plus to live somewhere we could swim, I know the water is usually too cold most places. But maybe an apartment with that amenity?
Places we've explored that seem to meet the criteria:
Spain, Portugal, Hungary and Poland, Greece.
We would LOVE your help! We need it! Thanks so much!