Start your Sheffield exploration with a visit to Kelham Island Museum, which showcases the city's industrial heritage. The museum has several interactive exhibits on the city's factories, including Sheffield's famous cutlery industry. After that, head to the Wetstone Gorge Trail for a scenic hike along the River Loxley. End the day with a pint at The Fat Cat, a historic pub that's been serving up real ale since the 1850s.
Day two starts with a visit to Sheffield Winter Garden, the largest city center temperate glasshouse in Europe. Next, explore Sheffield's contemporary art scene at the Site Gallery, showcasing a variety of exhibits from local and international artists. Walk over to Park Hill, an iconic complex of flats that were built in 1961, now converted into apartments and visitors center. End the day with a visit to one of the city's most famous landmarks, the Sheffield Cathedral that bears Gothic architecture.
If you have more time, you can explore Sheffield's green spaces and visit the Peak District National Park or see live performances at the Sheffield Theatres. If you have less time, you can combine the Kelham Island and Art and Architecture itinerary into one day and start with the Winter Garden and Site Gallery in the morning, followed by Park Hill in the afternoon and end the day with a visit to the Sheffield Cathedral.