Eastern Europe

Except for a large hotel bill or a large purchase that would be more than the cash you can withdraw, I see no reason to put anything on your credit card, especially as your card likely has a 3% foreign transaction fee.
ATMs are all over the place, and bank-owned ATMs charge no fee. Hopefully, you have a card with an enlightened bank or credit union here that also does not charge a fee (e.g., CapitalOne, or research this issue on this site, it's been done to death). Nothing can be easier than getting cash as you need it. Depending on your lodging costs, you can maneuver to pay those in cash also. It does not take too much planning to not have much excess cash as you leave a country.

Budapest: For the zoo go here http://www.zoobudapest.com/english/prices for prices and payment options. Excellent zoo by the way. This is the website for the Szechenyi Bath House: http://www.szechenyibath.com/prices-location.php but I don't know if they take credit cards. The question of how to deal with the money is a personal one. There is no definitive answer. From my point of view a 3% transaction fee is worth the convenience of using a card for dinner, etc. I have an Amex card that charges 1% and a Master Card that has no transaction charge. Both from Bank of America and both give excellent exchange rates. I also pull cash from ATM machines with no charges and also get excellent exchange rates. But the truth for an American in Europe is that the big bucks are on the hotel and the airfare which you are more likely than not going to charge. So even if you do get hit with a 3% charge on half the cost of the trip that's only 1.5% against the entire trip. You can save or spend more than that based on your in town dining and transportation decisions. I wouldn't sweat it. Do what feels good. I go to Budapest 2 or 3 times a year and I generally carry with me a fair sum of US Dollars mainly if I get stranded some place in transit; then I bring about 10,000 forints in small bills for the taxi and incidentals and I charge food and pull cash from the ATM for most everything else. You could just as easily show up at the airport with nothing but an ATM card or Euros or Dollars and either draw cash or exchange enough at the airport to get into town (as the rates at the airport are not great). There is some more information on currency here: http://budapestflat.shutterfly.com/fivedaysinbudapest look for a file called currency decisions and also download THE GUIDE for some general information on a lot of topics. Rick Steves has an article on the subject on the website.

Hungarian Forint Rule of thumb: REMOVE two zeros and divide by two to convert to dollars. Euro Rule of thumb: Add a third to convert to dollars. Metro Cards are the way to go in Budapest. The information on what they are, where to get them and how much they cost as well as how to use them are in THE GUIDE at http://budapestflat.shutterfly.com/fivedaysinbudapest Feel free to write if I can help.

Have you purchased Rick's books on Hungary, and the other countries? While they are not perfect (what is), they have answers to a lot of questions. You can also google "budapest ticket card". We were in Budapest last summer for 4 days. We bought the multiday transit pass, and it was a good deal. There are a lot of checks on the transit system, so you need either the pass or individual tickets.

Just returned from our third trip to BPest this year. Great as always. The trip reminded me of a few rules of thumb that I live by. Minimum Flight Connections 2 hours (2.5 preferred)
Minimum connection time when re entering the US 3 hours, except if you have Global Entry Pass in which case 2.5 hours minimum. On almost every trip there is at least one connection that pushes these rules to the limit. ON the most recent trip I would have missed two flights if I had shorter connection times. Travel time from Pest to the Budapest Airport 45 minutes. Leave your accomodations 2.25 hours before flight time - minimum.

My trip was Budapest to Vienna to Prague. I took out forints at the airport when I landed in Budapest - and got smaller bills from the foreign exchange clerk. I used whatever I had left over (leaving enough for cab fare and a snack) to pay down my hotel bill before I left. In Vienna, I had no problem - extra euros are just saved for my next trip. In Prague I used the ATM at the train station on arrival. My hotel was prepaid, so it was a little harder to manage not to have too many krona left over.

don't worry about getting cash - atms are everywhere. just to be sure you call your bank/credit card before you go. most places takes credit card anyways. make sure you buy metro tickets, there are people check frequently. euros work in some cases, i.e. if you stay with a family instead of hotels, then you can pay with local currency or euro. the budapest and prague train stations have plenty people renting rooms, and some are pretty good for the money.

Definitely let your bank and credit card companies know when and where you'll be traveling, so your debit/credit cards are not declined for suspicious activity. You may also want to have your daily withdrawal limit raised for your debit card.