You actually picked a pretty good town for getting around on public transportation. Everything with a few very minor exceptions is readily accessible on a tram or a short metro ride plus one bus.
I would love to help you, but a four day itinerary is a lot of work and will take some time. so when are you leaving?
You will need a copy of Eyewitness Guide to Budapest. The Top Ten guide will work and its easier to carry.
Here is a start. So your train arrives at 11am. Walk to the front of the station (towards the glass windows) and go down the stairs in the center. At the bottom on the right is a tourist office. By a 7 day travel card for each of you https://bkk.hu/en/budapest-7-day-travel-card/ Then go back up and at the top of the stairs look to the wall on your left.
You should see a door that says TAXI over it or near it. Anyway go out the door on that wall and it will take you to the taxi stand. Dont get in a taxi, call one; https://www.citytaxi.hu/ is my favorite. Remember call and reserve one. They will give you a number to look for at the stand.
You will be in your hotel at 11:30 more or less. So lets say your are out at 1pm. Lunch time. Walk out the door of the hotel and turn right. Go to the end of Andrassy ut (maybe a 3 min walk). Along the way you will see an OTP Bank. Use the ATM for Hungarian Forints. 25,000 would be a good start. If the door to the ATM is locked, swipe any card to get in.
Now at the end of Andrassy ut note the entrance to the M1 metro line (for future reference) then turn right and walk about 3 min more. You are at http://lugas-etterem.com/ great typical Hungarian food. After lunch return to that metro line entrance. Look across the street, see an identical entrance there? Yup, thats the one you want. Cross the street and get on. First stop is Opera exit. Get out look on the left side of the opera house and there should be a sign advertising the tours. The ticket office is near the sign. Go buy tickets for the 4pm tour. I am guessing a bit on this as I havent done the tour in about 6 years, I know I just keep seeing the sign there and the website says despite the renovations the tours go on. Now walk in the same direction the tram was going. Sorry, a bit of a walk but you need to see this. At the next major intersection, the Oktogon, you can get back on the M1 (make sure you use the station on the same side of the road that you got out or you will end up going the wrong way) Its all free and easy cause you bought TravelCards.
Ride the metro 2 stops, get off and walk to the next stop and get back on. Again, you gotta see it. Finally you get off again at Hősök tere (hero's square) metro station
You want to be back at he opera about 3:30 for the tour..... Walk to the metro station on the opposite side of the road as the one you exited and hop on. Get off at the Opera Metro Stop.
After the tour it is the metro again. This time you are heading for the end stop, Vorosmarty ter. Get off and look around a while as you are at the end of the big tourist shopping street Vaci utca.
Then walk down to the river. You are killing time till dinner!!
If the weather is good (no idea when you are going) there are a number of nice places to eat on the Danube Corso. Still too early, Right in front of you is the 2Tram, often cited as being the most picturesque tram trip in Europe. Ride it a few stop up and back.
That guide book I said to buy. Check every word i write against it. Everything is interesting but I am not going to point it out. For instance the metro you are riding is the oldest on the continent of Europe. Your hotel is on Andrassy ut the Champs-Élysées of Budapest.
See why this will take a while???
How do you feel about bath houses?
How late do you want to stay out
Interested in concerts and opera or theater?
What are your dates? Thats a big deal for a lot of reasons....