You could be forgiven for thinking that the 1975 probes Viking 1 and Viking 2 were the first missions to successful make it to Mars. In truth, the first attempts were made by the Soviet Union as early as 1960. There were 17 attempts prior to the first successful landing.
The Russians got there first in 1971 when the Mars 3 lander successfully achieved the first official touchdown. It made a 20 second transmission before going silent. The first image from the surface was disappointing, but it did take a wonderful image (shown below) from orbit.
Given the surprisingly large number of missions it is hard to keep track of them. Below is a wonderful zoom-able image showing all of the missions, including those that performed flybys.