Okay, nice location. In one direction is Andrassy ut which is nice and civilized. In the other direction is Kiraly utca which during the day is interesting but at night its the land of Spoiled-European-Trustfund-Brat-Party-Heaven.
Unfortunately, Google Maps won’t let you do multiple stops on public transportation so you will have to follow the segments.
First you will walk to the beginning of the 47/48/49 tram. Will take you about 5 minutes. The fastest way is to walk down Kiraly utca to Deak Ferenc ter where the tram ends its run. https://maps.app.goo.gl/TminXPeuC2cuHbWK8 Hop on and sit on the left.
You will take this all the way to the river. The trip is about 5 minutes, but the great thing is, because you have a travelcard, if you see something interesting, you can hop off and go look, then hop back on. Just show the travelcard if asked by and inspector.
Along the way you will pass, on the left, the Great Synagogue (Dohány Street Synagogue) and the Hungarian National Museum. Between the two, if you look out the right window you will see some wonderful antique books and art print stores. About 5 of them lined up just before you get to the Museum. Here is one. https://maps.app.goo.gl/iC6FYr2SNPkiG8kEA.
The tram continues across the river, but you want to get off at the last stop before the river. Just look at your google maps. If you go too far, you can always get on one going back.
At the stop before the river, you get off, and you are in front of the Great Market Hall. I don’t know what time it is, or what day of the week it is, but if it’s open, walk on in. Next, if you look towards the river, on the other side is the Gellert Hotel and Bath House (both closed for renovations but worth looking up for the history and the architecture). You will also see stairs right in front of you that go down under the bridge. So, walk on down. At the bottom you will find an entrance to the M4 Metro and a stop for the 2/2B/23 Tram line. Thats what you want. You want the tram that is moving from left to right. This is the tram line often referred to as the, or at least one of the, most beautiful rides in Europe https://maps.app.goo.gl/xzmGt9erJDEakavM7.
Sit on the left so you can see the river and the castle district of Buda and the Chain Bridge. Stay on for 3 stops, then get off at the Vigadó tér stop and walk up to DunaCorso Restaurant, sit down for a meal or a drink and suck in one of the most beautiful views in town. https://maps.app.goo.gl/18o2rKsPu3y4EEw88.
Now walk up the tracks to the next stop and reboard the tram in the same direction as before. We are off the Parliament building. The second largest in Europe I believe, but the most beautiful. Again, sit on the left. After a few stops you will find yourself at the first stop behind the Parliament building https://maps.app.goo.gl/8z7zvUdsAu9Zx6VZ7.
From here if you are interested and have time and energy, you walk down to the shoes on the Danube Memorial. Google Maps refuses to do it correctly, so I must tell you. You will walk directly from where you get off the tram, past the statue of the man on the horse and down the stairs to the river, the turn left and follow the river about 150m to the Shoes. https://maps.app.goo.gl/7ZAigtVtjN8N8ucdA
Now we take a nice walk across the back of Parliament and down a street lined with antique shops to where we find the next tram. https://maps.app.goo.gl/NNPY1BCb6mX4TVfz7 If you are interested in the antique shops, this is the one to go into. https://maps.app.goo.gl/9Lm3fAPweTjH7Rtq8