Any advice about day trips from Bucharest for independent travelers using public transportation?
Take the train to Brasov. Better yet, hire a guide for the day. Its pretty inexpensive in Romania. Send me a PM and I can send you a name.
Here is a link to a thread that was posted over 10 years ago, but you might want to look through it as the situation was similar, keeping in mind that it is an older thread.
https://community.ricksteves.com/travel-forum/to-the-east/bucharest-tips