Roman cooking braziers - Picture thanks to VROMA
Poor Romans in the countryside most often lived with their whole family in one room of a small apartment building. So they didn't have a separate kitchen. Instead, they cooked over a small fire or on a charcoal brazier, either in the courtyard or in their room (in bad weather). Poor Romans who lived in the city generally didn't have a courtyard, so they cooked on the brazier in their room, or they bought food in restaurants or from street vendors, already cooked.
A Roman oven from the House of the Stags,
Herculaneum - thanks to VROMA
Rich people had kitchens in their houses, but they didn't go cook in them themselves - they made their slaves cook dinner for them. Because of this, Roman kitchens were generally small and crowded, and not very nice, and in the back of the house where nobody would see them.
These kitchens had built-in clay ovens, with a sort of burner on top like our stoves, only heated by a charcoal fire inside them. And they had wooden cupboards, like ours, to keep the dishes and food in. They had potracks for the pots and pans.
Ancient Roman Homes, by Brian Williams (2002). Easy reading.
A Roman Villa: Inside Story, by Jacqueline Morley (American edition 1992). For kids, with lots of pictures.
Ancient Rome (Eyewitness Books), by Simon James (2004). Also for kids, with lots of great photographs.
The Roman Banquet : Images of Conviviality, by Katherine Dunbabin (2004). By a specialist, for interested adults. What Roman dinner parties were like, and how they were different from Greek ones.
Houses and Society in Pompeii and Herculaneum, by Andrew Wallace-Hadrill (1996). By a leading expert in ancient architecture.