The botanical garden is located just east of downtown (accessible via St George V Street) and is a great place to spend an hour or so wandering the landscaped green fields. It is open to the public and there is no charge. There is a nice hike from the botanical garden up the hill to the east, which offers a wonderful view of Roseau and the Caribbean Sea. The hike is of easy-to-moderate difficulty and will take a novice hiker perhaps 40 minutes one-way. The hike allows for up-close exploration of many native Caribbean plants, many of which are common house plants outside of the tropics. A small shrine is located at the top of the hill, near a look out.
South of Roseau is an area known as Champagne, a clever name meant to attract visitors. That said, there is no charge to visit the area and it is quite nice. The name comes from the mild underwater geothermal activity, which produces gas bubbles that float from the bottom of the sand to the surface - producing a wonderful tickling sensation on the body. The beach in this area (like around most of Dominica) is fairly nice. Expect to comfortably spend about 1-3 hours at Champagne. You can access this area by taxi, bus or private car and you will need to walk down from the road to the water.
Most natives know where it is. It will feel cool at first but it takes only 2-3 minutes to get used to it. The water in the pool feels very pure and soft against your skin. You could relax in there for hours.
Soccer games in Roseau are played on the local field. They occur fairly often and are a great popular spectator sport. There is a small entrance fee.