Wendy, the Wombat is in probably the best location of any hotel in town.
Kiraly utca (King Street) is typical Budapest street lined with local shops and restaurants. Also connecting to Kiraly utca, almost exactly to where you hotel is located is the Gozsdu Udvar. (Gozsdu Courtyard). The court yard is a series of interconnected building courtyards that connects Kiraly utca with Dobb utca and the heart of the old Jewish deportation ghetto. In the courtyard are a number of restaurants and bars. We eat breakfast at Café Vian's most mornings when in town. At the exit on Dobb utca is Spinoza Café which has a fixed dinner combined with a Klezmer concert on Friday nights.
Further up Kiraly utca are a few more restaurants and some design stores, and clothing stores. All local shops, no franchise businesses. On the left at the corner of Vasvari Pal utca you will find a wine bar called Kadarka. This is my evening hang out; especially in good weather when the outdoor tables are set up. Nothing fancy, just nice people and cheap wine and cheap pate.
Still further up, past the yellow church and also on the left, if you are having a hamburger fit (I get these when I am away from town for too long) is Zing Burger. Decent burgers for Europe, but for some reason they don't have catsup in the place. Fortunately there is a little market right across the street where you can purchase a small bottle.
Still further up is the back entrance to Liszt Ferenc tér (more food and drink) and the Liszt Academy of music. Worth going to for a concert or at least for the tour. Look on their web page, but I think the tours are at 11am weekdays. But check me on that.
Still further up you get to the ring road and the 4/6 tram which will take you to Margaret Island, but its sort of the long way.
Now if you go the other direction you are just a few feet from Deak Ferenc ter which is pretty much ground zero for seeing Budapest. From the square you can take the 47/49 metro to the Great Synagogue, the Great Market Hall and the Gellert Hotel.
You can walk across the street and be on Vaci utca in a matter of minutes. This is sort of the ground zero for tourist hype, but you have to see it just the same. A hundred feet from Vaci utca is the river front and the famous No2 Budapest Tram that will take you to the Parliament in one direction or the Great Market Hall in the other.
Also very near the entrance to your hotel is the end of Andrassy ut under which runs the Historic M1 metro line. Walking up Andrasy u you will find expensive shops, the Opera House, Embassies and at the end Hero's Square, City Park and Bath House. If you get tired, take the metro for certain sections of the trip.