National airlines fly in and out of Madang on a regular schedule, including jet aircraft.
Many "buses" (typically minivans or trucks where you sit in the back on a bench) connect to nearby towns. You can catch them at the central bus station near the market. Each vehicle has a number to indicate its destination. You can also flag down any passing vehicle with the right number. Vehicles are typically crowded. Passengers will be intrigued and interested if you join them this way if you're obviously visiting.
There are many boats visiting its port, so there are likely options for water travel also.
Many visitors come to Madang for the fine Scuba diving available in nearby waters. Others to see a pretty, small, relatively safe PNG, town that is a wonderful place to walk around and explore. The shoreline around the town has many small sandy swimming spots with grass, large trees and shade.
It is a great town to relax and recuperate. Large stores have most amenities the traveler will need, as well as a lot of local items that are interesting to see and try. Local residents are curious but reserved around travelers, but are usually friendly and helpful when approached.
In the center of Madang is a busy local market. People come daily from the highlands with produce to sell and load up with goods to take back with them at the end of the day. A wide variety of fruits and vegetables are available.
Outside of Madang are many small villages. Many of these offer tours and will show you local customs and their way of life. The people are extremely friendly and a tour is highly recommended. Any of the hotels in town can organise this for you for approx US$25/person for an afternoon including all transport. There are over 200 bats that take flight in the evening.
There isn't a lot to buy that you can take home with you. At least not typical tourist trinkets. In the villages they will offer you pottery to buy or local crafts which may be interesting to you.
There are not many restaurants outside of the few hotels who serve both guests and people passing by. The golf club serves meals and drinks though its weekday daytime hours can be limited. Some stores sell a limited selection of local 'fast' food. You may be able to find someone to sign you in as a guest to the members-only boat club that is a favorite of local business people and ex-pats.
The usual cautions apply for visitors eating in developing countries. Places catering to tourists are clean, the food is fresh and the water safe to drink. Seafood and produce are plentiful. Fresh coconuts are easy to buy and provide a refreshing drink. Ask someone for help, and they'll quickly brandish a large knife seemingly from nowhere and expertly cut it open.