Sheffield is a large city, and some suburbs are served better by the public transport system than others.
Probably the best and most comprehensive system is the bus system. Locals are friendly, and if you need help knowing where to get on or off, they'll invariably be willing to help.
The SuperTram, where it operates, is a great facility, but unfortunately does not cover many areas. It does, however, swing through most of the major areas such as Meadowhall (major shopping centre), City Centre (shopping, museums, many hotels), University of Sheffield, Hillsborough (major football venue), Sheffield Arena (concert venue and eateries).
Taxis operate in 2 forms - The minicab which must be preordered and cannot be hailed on the street, or the 'black cab' which are hailed on the street and those for hire are indicated by an orange light on the roof. Neither are a cheap alternative, but for short trips are very convenient. It is wise to know exactly where you want to go before getting into the cab and state your destination clearly to the driver. The odd unscrupulous cab driver may weave a circuitous route to your destination if one appears too uncertain!
Trains do not play a big role in transport within Sheffield, but are a good way of traveling to/from nearby towns.
Driving in Sheffield, like any city, can be a frustrating business, but in comparison to other large UK cities, is relatively painless (outside of rush hour). There are a few one way systems in the city centre, which are best avoided, but once into the suburbs it's not so bad. Narrow streets with heavy parking are common, and a courtesy system operates ie. Whoever can safely pull in first does. Those traveling uphill should take priority.
If you want to go on foot, be prepared for some hilly terrain! City centre is easily covered on foot as it is reasonably small, but if you're going further afield, consider other alternatives.