Nice walking trail to the Russian Memorial--a memorial erected in memory of the Russian sailors and Aleuts killed during the 1804 battle. It was once though that they were buried at the site of the memorial, but a careful inspection of the area found no evidence of graves. Still, it's a nice walk and there is a great view from the beach near the memorial.
Bring a raincoat--it rains a lot in Sitka. If you venture out onto the beach to fish for salmon, make sure you go out on the jetty until you are seaward of the fresh-saltwater markers. It's illegal to keep salmon caught upstream of the markers.