Erin, this works for the OP too, so we aren’t hijacking the thread yet. But if it goes much further you might want to start a separate thread.
Trip Reports: TexasTravelMom’s trip report is one of the best. Worth reading. The link is above.
For concerts maybe the Basilica concert as its probably going to be more convenient as its in Pest. https://www.bazilika.biz/en/the-basilica-s-choir Show up early and go to the top for the view.
For services, then maybe you do make the trip to Buda https://matyas-templom.hu/home Or for another look at life in Budapest think about one of the Synagogues for services. This was one of the largest Jewish communities in Europe before the holocaust and the Jewish influences are strong in town and the community is still strong.
I am no expert on the Bath Houses as I have only been to two and not that often, I do know that the Szechenyi Bath is easy and not intimidating for a first time experience. https://www.szechenyibath.hu/ again, convenient because its in Pest. If you are staying in Pest, its 10 to 20 minutes from everywhere. When I go, I rarely stay more than 2 hours. Get a cabana, bring your own towel (or buy a souvenir one from one of the vendors outside – I now own several) and flip flops. Nothing is timed.
For Szentendre take the H5 commuter train to Szentendre about noon. About a 40 minute trip. Have lunch at a place that looks out over the river, wander the streets then take the 5pm water taxi back to Budapest (about 45 minutes). https://mahartpassnave.hu/en/timetable/danube-bend-with-hydrofoils/budapest-szentendre-budapest Buda in the morning and Szentendre in the afternoon makes for a nice day.
Anniversary dinner? https://www.rezkakasbistro.hu/ or http://kiraly100.hu/en/king100restaurant/
June Events: Mister E sitting outside at Kadarka Wine Bar at 7pm (29 June, among many other nights). Sorry, June is a quiet month but there are so many good music venues (Liszt Academy, Budapest Operett, Hungarian House of Music, Opus Jazz Club, Budapest Jazz Club) that there is always something to do.
September Events: Jewish Summer Festival (3 - 10 September). Buda Wine Festival (7 – 10 September) and there is a Wine Lovers Event that I think will be at the Corinthia Hotel (16 September)