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.
The neigbourhood called "Spiegel" is beautifull. Furthermore there are some parks that are wortwhile.
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 recommanded.
Filmhuis, 't Elcker Theater. This is a cinema that shows only cultural movies.
Spant - a large theatre that also serves as a congress centre.
Stars - a club and to get in you need to be a member or you need to be introduced. You could give it a try though since it is the only place to really dance.
See a sports game. There are plenty of sportsclubs that often have good games during the weekend.
There is a good climbing hall in Bussum.
Las Brassas, Brinklaan 62. Serves decent Belgian, Dutch and French cuisine at reasonable prices.
El verano, Herenstraat 8-10, Bussum, ph: 035 6932960. Serves a decent meal for a reasonable price.
Pepi's Stationsweg 16-22, 1404 AN Bussum, ph: 035 6943724 (email:email@example.com fax: 035-6948470)  Serves good steak
Archibald Schimmelpenninck, Brinklaan 25, Bussum, ph: 035 6943030 (crossing Generaal de la Reylaan and Brinklaan) A grand cafe with a really nice atmosphere
Jan Tabak Amersfoortsestraatweg 27, 1401CV Bussum. Luxurious dinner and also a hotel.
Most pubs are around the train station. Around there is for example a popular cafe de peuk, next to de Jaargang and the irish pub Murphrey's. South of the centre there are two popular bars De Kraan and De Boemel. Non-youth could try in the mood which is in the very city center.
Golden tullip (Jan Tabak),
Hotel Bastion (near station bussum-zuid)