The only specialized equipment you will need is an orienteering compass. Most orienteering compasses have rectangular base plates under the compass dial. If you cannot obtain one, most clubs have a few rentals at each event.
Wear comfortable clothing appropriate to the season. Long pants are generally recommended at any time of the year to protect against vegetation and undesirable insects. Walking or running shoes are sufficient for beginners.
It's also a good idea to wear eyewear to protect against branches, a watch, as there is typically a time limit, and a whistle to call for help in case of emergency. In fact, whistles are required at many orienteering events for safety's sake.
Once you know that orienteering is the sport for you, specialized equipment and quick-drying clothing are available from a number of vendors. Competitive orienteers typically purchase:
In the U.S. and Canada, orienteering vendors often attend national-level "A" meets; otherwise, you can find them online.