Columbus was trying to find a faster route to India. While he was searching, he "discovered" the 'New World.'
Cabeza de Vaca
Cortez was exploring for gold, glory, and God for Spain. Along the way, Cortez conquered the Aztecs and became the first governor of New Spain, aka, Mexico.
Pineda was ordered by Jamaica to explore the new area for a faster route to the Pacific. Although he did not find a faster route to the Pacific, Pineda was responsible for making a map of the western coastlines.
De Vaca crashed his ship looking for gold & glory. He did, however, meet the Karawankas. He told stories about cities of gold, which led to more exploration.
Pineda was looking for the 7 cities of gold. Although he did not find them(because they didn't truly have gold), he did cross the sea of grass(panhandle).
La Salle was looking for the Mississippi river when he got lost and ended up in Texas. He got the French to claim Texas in it for two years.