If it's early September, you can still go to the beach both in Romania and Bulgaria. The Black Sea is not spectacular and it's pricey, but it's an option. (Depending on the type of scenary you prefer )

Outside Bucharest, some nice places I can recommend: Brasov, Sighisoara, Sinaia. I like the Bran Castle (known as Dracula's castle); it's a nice place to visit anbout 10-20 miles from Brasov. You have to get there by car as there is no train there. They may have some buses going there from Brasov if you don't want to rent one.

You could do something like... take the train to Sinaia, spend the day and the night there (go see the Peles castle, eat at the Serb's tavern, etc). See how you like it and if you want to spend more time there. Then head into Brasov. The center of the city is very nice. (Take a quick trip to Bran, maybe?) Maybe spend the night in Brasov and then take the train to Sighisoara. It's nice there. All throughout, definitely take the time to relax. If you try to go too fast you won't enjoy it.

While in Bucharest you might like White Horse (English bar with tons of expats). Bars aren't very popular in the summer; if it's nice out most people would rather go to a terrace (~beer garden). So if it's early september it might be tougher. If it's closer to the end of September then things are different.