Not sure exactly what advice I can give you in the way of meeting new friends.
I am a very, very young 62 year old married lady. Actually a very, very immature 62 year old.
I did meet a few very nice HU ladies at a gym I used to go to. The only thing was the ladies in my age group didn't speak English so I made friends with a couple of ladies that were too young for me, at least 20 years my jr.
They were also single ladies so that was another issue. We had them over for dinner a few times, went to their homes for dinner and drinks a few times but then I realized their lifestyles were too different then mine, the age thing was not that big of a deal.
One has since married and had a baby and the other was on the "gay" side? At least that's the vibe my husband picked up on when she brought her GF to my home, no judgement there but like I said the lifestyle was too different. Couldn't exactly see where that friendship would be going.
My single sister had many gay friends of both sexes though she was straight. It lead to a few problems when one lady took a shine to her too much.
I would think when the weather gets better you will find more people outside at the park and in the 5th district area, walking around, hard sometimes though because that area also has allot of tourists who will be leaving town.
I imagine you will slowly meet new friends though your job.
There is a ex-pat meeting club, forgot exactly what it is called, will post when it comes to me.
I met a American lady who happened to be like me in many ways , she was younger but she was married to a HU like I am, had children, didn't speak much Hungarian ,was a hairdresser like me and also raised in Cal. like me.
Sad, she moved out of Hungary. We met for coffee a few times but she left soon after meeting her.
We met at a large grocery store and exchanged phone numbers.
Have to be open but also remember most Hungarians are a bit guarded and it can be hard to get to know them fast.
If you like sports, I know many people do jog at Margaret Island , they also have a few swimming pools there as well.
I also ran into a fellow American women a few years back who has lived in HU for now over 15 years. She was much younger then me too. Met her while she was dog walking her dane.
It is easy for me to meet people, I often have to check myself not to be overly friendly as it doesn't fit in my lifestyle these days, retired with a husband who likes be alone allot with just me, simple and quite no drama.
I would say be friendly at work, let people know you are open to meet interesting nice people who have some of the same interests you have.
Going to ruin bars and drinking can be fun but I seriously doubt you will meet anyone who doesn't have an issue or two. I know, had a single sister who made bar friends...
Look online for ex-pat clubs.
When the weather gets warmer there will be tons of people out and about the city. Maybe buy a bicycle and ride?
Maybe another ex-pat on this site will be game for a walk and talk?