The GMPTE (Greater Manchester Public Transport Executive) website includes timetables for the main bus services for the area around Cranswick Square and to and from Moss Side. These are:
GMPTE has a journey planner here for more complicated journeys, including trains and trams as well as buses.
Manchester Community Transport also runs the Local Link service, which uses minibuses to take people door-to-door. To use this you need to contact them for a user number, and then you can book transport within the Moss Side/Hulme area by phone or online. More details here. This is especially useful for older people, people with disabilities, people with young children or people with heavy loads to carry, for whom normal public transport may involve too much walking and for whom a private taxi is too expensive. The Ring & Ride scheme provides a similar service, but covers a larger area and is cheaper than Local Link. It also runs occasional special trips to the Yorkshire Dales and other day destinations around Manchester.