The Ulster Museum in the Botanic Gardens is Northern Ireland's main national museum, with a comprehensive and free collection covering art, history, and natural science. The history section covers the full sweep of Irish and Ulster history from prehistoric times to the present, including a thoughtful and balanced account of the Troubles that contextualizes what you see in the city.
The art collection is particularly strong in Irish painting from the 18th to 20th centuries, and there's a fine collection of archaeological objects including the Malone Hoard — a cache of Neolithic stone axes found in Belfast in 1865. Allow at least two hours. The building has been sympathetically expanded and is pleasant to navigate.