Like all Vietnamese cities roads are dominated by motor bikes and these are the cheapest way of getting around, if you simply stand still beside any road bikes will zoom up and ask where you want to go.... fix the price first! There are metered taxis and for short distances bike rickshaws. Most places to stay will offer advise on prices for trips and even arrange transfers and trips for you at very little extra cost. If you are taking bikes around the city make sure you have plenty small change for the fare as you can be sure they will not have change!
The main reason for visiting Hue is the ancient city buildings, the main two are the Emperors palace in the forbidden city and the emporers tomb (outside the city) but there are pagodas and temples all over the city. You have to pay an entry fee, there is a single monument ticket valid for same day use only, or a 3 site and a 4 site ticket that is valid for two days, so if you are going to visit more than just the forbidden city get the multiple ticket. Prices above are for 2015 and in Vietnamese Dong.
The largest monument to explore is the Forbidden City and you could easily spend all day wandering around the many gates, temples, pagodas, gardens and pathways imagining what life must have been like in times past, of course if you had been caught there in those times the penalty was death, nowadays its just a pair of sore feet, there is a lot of walking to be done so wear comfortable shoes and take plenty water. For those who want to save thier feet you can take the tour around the site on a guided golf buggy.