Most homebrewers start off brewing in their kitchens on their own stoves. This is perfectly safe and should cause you no concern except to be careful about boilovers in your brew kettle, which can be difficult to clean off of certain types of stovetops.
However, usually only a partial boil can be performed on a stove top. Many brewers eventually move to doing a full-wort boil on a powerful propane burner. Propane burners are not intended for indoor use, and should never be used indoors, even in an area that you think has sufficient ventilation.
