Candle-lighting
This site shows Hebcal's standard for Israel, 20 minutes before sunset. Jerusalem's custom is 40 minutes, and some families and shuls in Mevaseret follow it; ask your shul. See Shabbat times.
Shuls
Mevaseret has many Orthodox congregations; OpenStreetMap lists among others Beit Yakov, Achdut Israel, Yeshurun, Hazon Yechezkel and Mishkan HaImahot Rachel VeLeah. Kehillat Mevasseret Zion, a Reform congregation, was founded in 1993. Har Adar has the Harel synagogue. See every shul listing; davening times are not listed yet.
Mikvaot
The religious council runs two mikvaot in Mevaseret, HaBustanim (in the Castel area) and Sde Hemed, and there is a women's mikveh in Kiryat Ye'arim. Call ahead for hours. See mikvaot.