- If we have 10-12 days starting in mid May, where do you suggest?
Too broad a question as we all have different favorites but as others have suggested, stick to one country. I'll go with Italy as it's my fave, is not difficult to travel, and the "Holy Trinity" (Rome, Florence, Venice) is a good plan for first-timers with the amount of time you have.
- What city should we fly into to start the journey? Should we choose the same city to fly out of?
To save time-wasting backtracking, you want to fly into one city and out of another. Venice>Florence> Rome is the usual recommendation as international flights out of Venice can be uncomfortably early in the morning.
- Should we try to plan the number of nights in each place so we have a hotel?
Absolutely. May is High Season in much of Italy and other cities in Europe and you're already a little late in the game for booking. You also do not want to waste time wandering around a strange city with your luggage looking for a place to stay. Pre-booking allows you to know precisely where you will be, how much it will cost, and what to expect as far as facilities, such as an elevator (those can be rarer than not), breakfast and whatnot.
- Should we try and plan a night train ?
No, at least not for the 3 cities mentioned as they are not far enough apart to need them, and I wouldn't recommend it for further-flung locations either for reasons already listed.
- What should we make sure to see?
Get a guidebook and spend some time with it. It's THE best way to make your list as everyone's interests are different. In fact, a guidebook will help you with some of your other questions.
- What kind of budget should I plan for?
Another too-broad question. It largely depends on how much you want to spend per night for accommodations, buying advance train tickets per on-the spot, what you want to spend on meals, etc.
Planning a trip involves quite a lot of homework so you learn where to stay, how to get around, how to save money (if that's a consideration), how to GET money abraod, and generally how things work. Homework of this sort doesn't appeal to everyone so if you find putting an indie plan together isn't for you, you might want to consider a package tour that takes care of the bulk of the details for you. My husband I are independent travelers (so far) so we haven't taken a tour but this Venice>Florence> Rome combo has been very positively reviewed among people who've done it.
https://www.ricksteves.com/tours/italy/venice-florence-rome
If Italy isn't top on your interest list, then take a glance at the tour offerings for other countries.
https://www.ricksteves.com/tours