Enjoy Budapest - it is one of my favorite cities.

Looking at a day trip to Vienna is a tough one... I have
taken the train between the cities, but to do both ways
as a day trip is too much, in my opinion. I don't think
Vienna is the kind of city where you can see a couple of
the big sights in a couple of hours and get a sense of
the place.

I might suggest two other choices, both designed to
give you more exposure to Hungarian life.

An easy trip is to the area known as the Danube Bend -
Szentendre is the name of the town (I hope I am spelling
that correctly). The trip is very easy - it is a train, but it
is called the HEV, and is basically a commuter train. I
think the trip takes an hour, and it gives you a chance to
walk around some of the older parts of the city. It is a
bit touristy, but still one of the best choices for a day trip
in my opinion.

If you want something a bit different, you might also think
about taking a train to a Hungarian city of Debrecen. I don't
know how long the trip takes, but Debrecen is one of the
larger cities in Hungary (although any city other than Budapest
is much, much smaller). Some universities in Debrecen, and
it might be a way to get a better feel of how the Hungarians
spend their days when not living in their massive national
capital.

Enjoy your trip!