There is a range of cheap or slightly higher-end places to eat in Addis Ababa, from cafes serving pizzas and other Italian cuisine, to burger joints, converted colonial mansions and Indian/Chinese restaurants. A meal at a local Ethiopian restaurant is also thoroughly recommended--it is an experience! Guests will be given soap, water and a towel to wash their hands before pieces of injera (type of bread) are used to scoop up wat (either meat, chicken or vegetables in a hot pepper sauce) or tibs (fried pieces of steak). No cutlery is used. There is also a big selection of seafood available in Ethiopia. It is a good idea to avoid salads-- these can contain high levels of pesticide and germs for liver fluke.
There is a decent variety of nightclubs and discos in Addis Ababa (including Bar Shamas, 3M Club and the Ghion nightclub which features a great Ethiopian band), cinema houses showing mainly American, Indian and Arab films, and some good theatre showcasing anything from Shakespeare to Ethiopian traditional music and dance.