In Of Mice and Men, Lennie Small's main goal is to one day buy some land with George, set up a farm, and live off the "fat of the land." In order to achieve this dream, Lennie is just smart enough to follow George, who claims to know exactly how to go about it. However "smart" this may be, Lennie believes this is his best shot to live a peaceful life, away from the mean-spirited tendencies of the typical, loner ranch hands. So, indirectly, one of Lennie's motivations is to keep George happy in order to achieve this dream.
Lennie is also motivated by his instinctual, animal desires. Since he basically has the mind of a child, he doesn't know when to restrain from these baser instincts. His intentions are always loving and curious but his brute strength gets him into trouble. He can't stand not to pet the mouse or grab a dress, but the results end up in disaster.
Since Lennie is always in need of someone to take care of him, he knows firsthand what it is to care for someong. The whole thing with petting animals is not just an instinctual or childlike response. This is his way of taking care of something more simple than himself. Being that one of his future dreams is the farm, where George has conditionally promised that Lennie can tend to the rabbits, another of Lennie's secondary motivations is that this dream will allow him to take care of something, to assume some responsibility, some sort of adult role. This goes along with his generous, innocent personality. Lennie wants to give care as much as he's received it.