gajiti

1. (of plants) to grow; to cultivate
   Baka u bašti gaji paradajz i papriku. — Grandma grows tomatoes and peppers in the garden.
2. (of animals, children) to raise; to breed; to keep
   Na selu gajimo kokoške i dve krave. — In the village we keep chickens and two cows.
   Sama je gajila troje dece. — She raised three children on her own.
3. (of feelings) to harbor; to cherish
   Još uvek gaji nadu da će se on vratiti. — She still cherishes the hope that he'll come back.
   Gaji duboka osećanja prema njoj. — He harbors deep feelings for her.
4. (of tradition, culture) to nurture; to maintain
   Naša porodica gaji stare običaje za Božić. — Our family keeps up the old Christmas customs.

For bringing up children in the sense of upbringing, "vaspitavati" or "odgajati" is usually used.