Dada in Surrealism
You can think of Dadaism as the older cousin of Surrealism. The Dadaists, like the Surrealists, were way into in the irrational, and their work emphasized this. They also wanted to destroy the old cultural forms and create new ones.
A big name in the movement was the French artist Marcel Duchamp, who coined the term "anti-art." He created works of art out of ready-made objects (like spades and, ugh, urinals). The Dadaists ignored traditional artistic conventions; in fact, they loved breaking them.
But Dada was different from Surrealism in a number of ways. For one thing, the Dadaist movement didn't have a "leader" in the way that the Surrealist movement did (it was led by André Breton). Surrealism was based in Paris and was very closely associated with the city, whereas Dadaist artists and writers were scattered all over Europe. But the two movements had a lot in common, and Dada served as a big inspiration for the Surrealist writers based in Paris.
Chew On This
Tristan Tzara is a poet who is associated both with Dada and Surrealism. His "Poem for a Dress" focuses on irrational imagery—and of course the Dadaists were big on the irrational.
Tzara's poem "Way" is another example of Dada writing. Have a look at its striking imagery here.