Only 2 full days...

With two days, yes it is crazy. Try to see as much as you can in Budapest and you will still see only very little fraction.

My first impression would be that the Vienna trip would be crazy, but maybe you have a good reason? Two full days means you are arriving one evening then spend all the next day and all the day following and leaving early the following morning. Give me the dates and times, where you are staying and i will give you some ideas. March is actually a very, very good time to be in Budapest and if one of the days is March 15 that is even better.

With 2 days a night cruise and walking about and seeing the interior of Parliment building is about it.
you will enjoy this walkable city. Hopfully the weather agrees. Enjoy.

With such a short time, I would stay in Budapest. To maximize your sightseeing time, get tickets to an evening concert at St. Stephen's instead of a daytime visit. Get there early and you'll have time to walk around the interior a bit and take photos. The lighting is better after dark, so you'll see much more detail than in daytime though you won't see as much of the church.

The train ride is at least 2.5 hours each way. Then allow time to/from station to the city center (4 times, twice in each direction, twice in each city) - probably much more than an hour. That's 6-7 hours just back and forth. You'll barely have time to ride the Ring Tram and grab lunch in Vienna.

We just did something similar to this...against my better judgement we stayed for only one night in Budapest with two full days on either side. I regret it terribly! I had no idea there was so much to do and see and I plan to go back for a week at minimum. We had time for a hoho tour, dinner, a river cruise at night (a MUST do in this city), a walk through the park the next day, and a visit to St. Michael's church on top of the hill overlooking the city. That's it. We missed so much of what the city has to offer...it's extremely beautiful.
It's too far to Vienna for just one day - you will completely short-change both of these amazing cities. Stick to Budapest, see what you can, and plan to go back to both Budapest and Vienna on another trip.

If you do want to go to Vienna here is an option. Since you said you will have 2 full days I am assuming you are arriving one evening. Take a train to Vienna that evening. You can get there in 3 hours plus or minus about 10 minutes. If you are able to get there by 6pm you can still have a pretty good evening in Vienna. There is a train almost every hour. Get a room on the Kärntner Straße so you will be well centered for the short stay; maybe the Astoria or the Sacher. Find a train back to Budapest about 2 pm the next day. That gives you the morning to see Vienna and lunch in Vienna and will allow you to enjoy the evening in Budapest. Would I do it? No. I have spent years trying to see Budapest and I have only scratched the surface.

The introduction to Rick's Budapest guidebook (pages 26-7) proposes a rough plan for spending two days in the city. Each day, you can usually fit a couple of must-see sites that you've prioritized, as well as some flexible time to wander and make discoveries. A local guide can also help you to be efficient with your first morning or afternoon. The sightseeing overview at http://www.ricksteves.com/europe/hungary/budapest should convince you that Budapest will keep you busy.

You can have a great 2 days, especially if you get the night before, but too much depends on the day of the week and more depends on the actual dates so without that information i don't want to take the time shooting into the wind.

Thank you everyone on your input! I tend to agree with all of you about going to Vienna... just too much. Chani, your idea about the concert is fabulous! 2 birds with one stone! James, we are staying in the Ferencvarcos District, close to the Holocaust Museum. Our full days are on a Friday and Saturday. We do arrive the night before, so we are going to hit the pavement running on Friday to see as much as possible. We leave for Prague on Sunday. I would love to do a river cruise, but concerned about the weather. That being said, I've read both opinions on getting tickets for a cruise before you leave home and buying the tickets there. What do you think?
Thank you again!

So, I saw Worldinbetween's post and tried to figure out how she put you so close to the market. She's right, great market, but you are half way across town from it. I also screwed up and had you a little further north than you will be staying. I shouldn't try to do things by memory. I made a few corrections below in BOLD

Arrival Thursday 3, 10, 17, 24

Means you miss the 15th which is a real shame because it is a National Holiday with all sorts of celebrations around town.

Depending on what time you arrive you might stroll out for a good dinner. You hotel is sort of “out there” and you haven’t got a lot of time. So find your way to the Corvin-negyed M3 metro station use the ticket machine to purchase 10 tickets and take the M3 to Deak Ferenc ter. From here it is less than a 10 minute walk to the heart of Danube Embankment. Find a place to eat and enjoy the view. You can walk the trip if you want. Shouldn’t take more than 30 minutes . https://onedrive.live.com/redir?resid=C9F4C533EB93D92E!16627&authkey=!AFFXgdA2pX_hJew&v=3&ithint=photo%2cJPG Worth the walk or the subway and the tram.

Friday find your way back to the M3 station at Corvin-negyed from there on the M3 one stop to Kalvin ter and from there the 47/49 Tram to the Great Market Hall. Follow your nose in the basement. Afterwards take the 47/49 Tram across the river to the first stop. It’s the Gellert Hotel (google along with Eichman and Schindler), the get on the 19/41 Tram to the funicular and see Castle Hill (3 hours more or less – use a guide book). Then walk down the hill and walk across the Chain bridge, find the 2 Tram back to Fovam ter and then the 47/49 tram to the Great Synagogue and the old Jewish district. A couple of amazing synagogues here along with some pretty interesting local streets with restaurants, clubs, ancient unrestored architecture. Probably has as much character as any place in town. A paid walking tour is always good in this area. Take the Great Synagogue tour.

For dinner book a table for the Spinoza Dinner and Klezmer Band show (only on Friday night). Google Klezmer. From where you are staying……..take a taxi. Find a nice wine bar for after dinner. My favorite is Kadarka; not too far away.

Saturday I would go back to Corvin-negyed and hop on the 4/6 Tram to the New York Palace for your morning coffee. All of these places you need to google and read the history. Then back on the tram to the Oktogon. From the Oktogon you will walk up Andrassy ut enjoying the architecture and mansions until you reach Hero’s Square (google), from there into city Park to the Bath House. If nothing else to see the architecture. Then back on the M1 metro to the Oktogon (google the M1). From the Oktogon walk down Andrassy to the Opera, shops, stores. At the end of Andrassy is a Georgian Restaurant. Have lunch. Now to the Basilica. Climb to the top for the view and pay the 300 ft to see the sacred hand. Now down the pedestrian walkway about halfway to Oktober 6 utca which will lead you to the Szabadság tér, Ronald Reagan, the US Embassy and then to the Parliament Building. From the Parliament you can take the #2 Tram back to familiar territory. Pretty much a day.

Dinner river cruise with tickets you purchased on line.

Wow! Thank you James!!! Great information! I will definitely look up all the places you said to see. One more question... we were also wondering about the bath houses. Is there one that is better then another one? The Hotel Gellert? My friend and I want to go, but I'm a bit "iffy" about it. Thank you again!

Bath houses. They range in character from 16th century Turkish to Baroque to Neo Classical. Each has its own attraction. While I know its the most touristy (half of the people there will be tourists, the other half local) I just have fun at the Széchenyi Baths. Here, you choose: http://visitbudapest.travel/activities/budapest-baths/ If you are into such things then its just the thing to do.