The Arab Museum of Modern Art (MATHAF) holds a fair sized permanent collection of Arabic and Islamic modern and contemporary art works, along with changing special exhibitions. Permanent exhibits are free, as are seeing the sculptures outside, while special exhibitions are fairly cheap.
The museum is located east of the city centre, at Education City. That would normally make it a bit of faff to get to... Except... There's a free shuttle bus!
The bus (a bright pink minibus) runs between the Museum of Islamic Art on the Corniche and MATHAF every hour Wednesday to Sunday, timings on the MATHAF website. It's supposed to take about 30 minutes, but often takes closer to 45 in traffic.
There's some very interesting and good stuff at MATHAF, it's worth the bus ride out from the city centre!
(Supposedly nearby is also the gallery at VCU Qatar (Virginia Commonwealth University in Qatar). However, because of ongoing works for the Education City tram, you'll face a ~30 minute unsignposted walk through what's basically a building site to get there, and the same back, assuming a passing expat doesn't take pity and offer you a lift. Until the tarms are done, I'd say it's best to skip the small VCUQatar gallery, as it isn't an easy or fun walk)