There are a number of places to visit that are around or close to Annaba.
Seraidi: Seraidi is a small village up high in the mountains that surround Annaba. It's about 13 km from downtown. It's a great place to enjoy a day away from the hustle and bustle of the city and escape enjoying the tranquility and clean, crisp air. There is a famous hotel there known to the locals as "Hotel Al-Hawa", literally "hotel of the wind". It is called that because it's so high up that it breaks the clouds and offers a great place to stay and has amazing views. On a clear day one can see the amazing views of the coastline and natural beaches below. On a cloudy day there is noting but clouds. It's a great place to see the sunrise or sunset.
El Caala: El Caala is a small town about an hour and a half drive east of Annaba. It's on the road to Tuinisia so it is very service oriented. There are many hotels, clubs, restaurants to enjoy. The beach of course, and many souvenirs one can get from there. Not far from El Caala there is "La Vielle Calle" or "Old Caala". There is a wonderful beach there and ruins of a French fort which are beautiful. La Vielle Calle is about 30-45 minutes away.
Tunisia: Annaba is in the east of Algeria, not far from the Tunisian border. A drive to the Tunisian capital, Tunis, takes about four hours. The roads leading to Tunisia from Annaba are very well serviced and have numerous service oriented places on the way, such as reataurants, hotels and gas stations.