Councillors confirmed the closures alongside increasing the cost of adult social care services and cutting back on street lighting during the night.
Twelve paddling pools are to be closed over the summer but a decision on the future of leisure services was deferred.
The council needs to make ??70m budget savings over four years.
Although the Muni is to close, the Coliseum in Aberdare and the Park and Dare in Treorchy will be retained, which the council says is more than many other local authorities operate.
And while the Cynon Valley museum will close with , the Rhondda Heritage Park will be retained and developed into a heritage hub.
Bus route subsidies for services that are not commercially viable are to be nearly halved to ??0.4m a year.
Councillors also voted for part-night lighting of all streetlights in non residential areas and part-night lighting of alternate streetlights in residential areas.
Paul Cannon, deputy Rhondda Cynon Taf council leader, said: "There is no easy solution in these tough financial circumstances.
"Our funding has been dramatically reduced by the government in London, and this will mean that difficult and unfortunately unpopular decisions were necessary today and will be required in the future.
"It must be remembered, however, that even after these tough decisions, the level of provision offered in Rhondda Cynon Taf still compares favourably with other local authorities."
The Muni Arts Centre in Pontypridd and the Cynon Valley Museum are to close as part of budget cuts in Rhondda Cynon Taf.