We are headed to Budapest in July 2018 and have 10 days to work with. I'd love opinions about whether we should take the train to see Vienna and Prague while we're there, or should we stay in Hungary and rent a car to visit the wine regions. We are wine lovers and have visited California wine country a few times. Are the wineries in Hungary similar in offering tasting rooms and tours?
Take the wine tour since you love wine.
Also Vienna and Prague are hot and crowded with tourists in July.
Ten days isn't enough time to do Budapest, Vienna and Prague. Its barely enough for two of the three. If you are interested in the wine country, this is the premier wine tour company: http://tastehungary.com/tour/wine-day-tour/ which I think makes more sense then trying to do it solo.
There is plenty to keep you busy in Vienna and Budapest. Bratislava is a great place to stop and recharge your batteries before going into Vienna 39 miles to the north.