Ask your spouse, or a friend, or your dad to go with you. It does increase the cost, but it is a relief to have someone there who can help you over the inevitable rocky times, the crying jags, the jet lag, etc. Being alone in a foreign country with a small child can be overwhelming, and the feeling of isolation can be tough. Generally, the restaurants are disappointing, and tend to serve pizza, pork dishes, and odd sandwiches made with some kind of meat, french fries, mustard, pickles and ketchup rolled in a pita bread. It is hard to find traditional Romanian food, but insist that you be taken to at least one true Romanian restaurant, like Caru Cu Bere.
Taxis are everywhere and reasonably affordable.
We dressed casually most of the time, and comfortable shoes are a must for city exploration. We spent hours each day roaming around the city. We were never harassed, or stared at, and the people tend to be reserved but polite. Most college students and younger Romanians are very friendly, and will stop and engage you in conversation if they hear you speaking English. A good guidebook is invaluable, but don't believe what they say about bringing dollar bills and cigarettes for tips and bribes.