Candle-lighting
This site shows Hebcal's standard for Israel: candles 20 minutes before sunset. If your shul publishes another time, follow your shul. See Shabbat times.
Shuls
According to an English-speaking oleh writing for Nefesh B'Nefesh, most of Be'er Sheva's shuls are Sephardi, mainly Moroccan, and English speakers tend to gather in Mizrachi or Young Israel shuls; there is also a Chassidic shul in the Old City.
OpenStreetMap lists, among others, the Heichal Shlomo Synagogue in Dalet, the Eliyahu Haltzi Central Synagogue in the centre, the Struma Synagogue in Alef and Kehilat Eshel Avraham in Tet. See every shul listing. Davening times and nusach are not listed yet: ask the gabbai.
Mikvaot
The religious council runs 15 mikvaot, listed by the Ministry of Religious Services, in Alef, Gimel, Dalet (two), Hey, Yud-Alef, Kalaniot, Neot Lon, Sigaliot, Ramot (two) and elsewhere. Hours differ between summer and winter, so call ahead. See mikveh listings.