I suspect that most of the taxi services are good. I use City because it was recommended to me 20 years ago and over the years they have never let me down. And on a few occassions they have gone above and beyond. So, once I find something that works I dont explore much. https://citytaxi.hu/ We have Bolt too if you are more familiar with them. But no Uber here.
With the City Taxi App you dont have to know the language, its all input. The map tells City where you are and the map tells you where the taxi is. Pretty swift and clean.
Since you will be in Budapest a few days, I suggest you visit the New York Cafe. But come early in the day and during the week when there are fewer tourists.
https://newyorkcafe.hu/en/category/history/
https://en.mandadb.hu/cikk/1182839/Historical_guided_tour_in_the_Most_Beautiful_Cafe_of_the_World
Part of the reason for the suggestion is how you will get there.
You will walk out your flat and board the 4 or 6 tram.
Which way? South East of course. If its the 4 tram you get on look for one that says Újbuda-központ and if it is the 6 tram, look for one that says Móricz Zsigmond körtér. Those are the end stops and its how you tell direction on all of the public transportation.
If you get on going the wrong way the first stop will be the Eiffel designed Nyugati Train Station. The outside was recently renovated. Beautiful. Interesting too, if you want to to take a look inside. Then back on the tram going the right direction.
From the Oktogon you will go to the thrid stop called Blaja Luja ter. Once off, look up the intersecting street. See that beautiful old building at the end of the road? Thats Keleti Train Station and its why I suggested visiting the New York Cafe. Turn around a few times and you will see stairs going down beneath the street. Those go to the M2 Metro station which goes to the underground of the train station. How easy.
Okay, now the NY Cafe. Cross the street that the tram came down and walk back about 200 meters. You are there.
Afterwards if you want to go to go to other sights the majority will require you to first reach Deak Ferenc ter. Thats good too, buecause the best way will be that M2 metro line and once you see how easy that is, you will have no fear with getthing to Keleti your last day ............. or call a taxi.