
The neigbourhood called "Spiegel" is beautiful. Furthermore there are some parks that are worthwhile. Have a walk on the meadows. Go to Naarden and see the fort. On of the best ways to see the country side around Bussum is by bike. A bike tour through 't Gooi is recommended.
Bussum is easily reached by train: Naarden-Bussum station is 21 minutes from Amsterdam Central, and 23 minutes from Utrecht via Hilversum. The centre of town is about 5-10 minutes walk from the station.
You can cycle from the centre of Amsterdam to the centre of Bussum, at an easy pace, in less than 2 hours. Most of the route has separate cycle lanes.
Bussum is easilly reached by car. The A1 passes all around the town.
Use a bike or walk. A car will do fine too.
Most pubs are around the train station.