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If you're into hiking, there's a trail that goes round the outside of the city. The trail is 42 km, so it takes at least 2 days, and can even be broken up to 5 days, as it has 5 sections. Section 1 goes from the Negev Brigade Memorial through hilly terrain around Ramot neighborhood to the northern entrance to the city (10 km). Section 2 goes around the northern neighborhoods to Road 25 (7 km). Section 3 crosses Hatzerim Park (5 km). Section 4 goes along the Beersheba Promenade on the northern bank of the (usually dry) Beersheba River to Abraham's Well (8 km). Section 5 goes through the Beersheba River Park back to the Negev Brigade Memorial (9 km).
The Old Town was planned by Turkish and German engineers at the end of the 19th century. Its streets form a grid (very uncommon in the region). Many buildings date from Turkish times and have secret gardens behind high walls. Rehabilitated Smilansky Street has the best preserved buildings. Additionally there is a wide array of cheap stores and excellent ethnic restaurants and bars.
Located at the edge of the Old Town and on the Wadi Beer Sheva, this small site contains the well where according to tradition Abraham made the oath with Abimelech.
Located between the Old Town and the central bus station is a vibrant market for any kind of merchandize.
Located in the residence of the Turkish governor in the Old Town, this small museum contains an art collection and has changing exhibits.
An old train station, built in 1915 by the Turks, featuring steam engine #70414 from the 1930s. You can walk around the station freely and eat at a restaurant. There are also paid tours and a small museum.
Located near Kibbutz Hatserim, 5 km to the west of Beer Sheva. The museum has a collection of ~150 historical airplanes of the Israeli Air Force (and a few foreign or enemy planes). Some of the planes you can climb into. You can arrange for tours in Hebrew or English. To get here, take bus 40 from the central bus station (about hourly).