What difficulties do Lennie and George face throughout their adventures?

One is made aware from the outset not only of the contrast in size between George and Lennie but also of their obvious difference in intellectual capacity. Both these disparities are responsible for some of the difficulties they encounter throughout the novel. Since Lennie is so large and very childlike, he does not realize his own strength. In addition, his mental ability affects his attitude and approach to things. Lennie, like a small child, loves stroking soft things. He seems to have developed an obsession to explore, in a tactile sense, the texture of soft materials and objects, be they alive or dead. He derives tremendous satisfaction from fondling velvety, furry, plush objects. It this fixation that often gets them in trouble. 
Lennie's fetish creates friction in his relationship with George and leads to conflict between them. In chapter one, for example, George discovers that Lennie had been fondling a dead mouse. He is upset about it and scolds him.

"Give it here!" Lennie's closed hand slowly obeyed. George took the mouse and threw it across the pool to the other side, among the brush. "What you want of a dead mouse, anyways?" "I could pet it with my thumb while we walked along," said Lennie. "Well, you ain't petting no mice while you walk with me."

Another difficulty is the fact that Lennie hardly ever remembers what George tells him. George is upset that Lennie can't remember where they are going.

"Where we goin', George?" The little man jerked down the brim of his hat and scowled over at Lennie. "So you forgot that awready, did you? I gotta tell you again, do I? Jesus Christ, you're a crazy bastard!"

Since Lennie has such a poor memory, it complicates their situation; George has to continuously remind him of what to do and what not to do so that they can stay out of trouble. It is tragic and tough that at the end of the novel Lennie forgets one particular incident when they were in desperate trouble. When they were working a ranch in Weed, Lennie fondled a young woman's dress, and when she became upset and started screaming, he held on tighter. She became desperate and wrenched away from him, tearing her dress in the process. She claimed that Lennie had abused her and George and Lennie had to hide in a ditch for hours to escape their persecutors. In the end, Lennie fondles Curley's wife's hair, and when she tries to scream, he unfortunately breaks her neck.
A minor difficulty that the two experience at the beginning of the novel is when the uncaring bus driver drops them quite a distance from the ranch that they are supposed to work at. As George says:

"We could just as well of rode clear to the ranch if that bastard bus driver knew what he was talkin' about. 'Jes' a little stretch down the highway,' he says. 'Jes' a little stretch.' God damn near four miles, that's what it was! Didn't wanta stop at the ranch gate, that's what. Too God damn lazy to pull up.

This gets them into trouble with the ranch owner. They are, however, fortunate that he believes their story, although he is still suspicious, saying,

"All right. But don't try to put nothing over, 'cause you can't get away with nothing. I seen wise guys before..."    

Another difficulty the two men encounter is Lennie's interest in Curley's wife. He finds her "purty" when he sees her the first time. George has heard from Candy that she tends to flirt with all the men on the ranch, and he immediately assumes the worst of her. He tells Lennie to stay away, probably because he is thinking about the incident in Weed. Furthermore, Curley, the ranch owner's son, later picks a fight with Lennie and hits him in the face. Lennie breaks his hand, and Curley is humiliated. He resents Lennie, but Slim tells him not to talk about the incident because, he warns, he will tell the truth about how Curley came to be so badly injured. Curley, who is insecure, accedes because he does not want to be embarrassed. Slim's intervention probably saves the two from being dismissed.
In spite of all the difficulties they have, the two men share a deep bond. They are tied to each other because George has made a solemn pledge to Lennie's aunt Clara that he will look after Lennie. He enjoys the task of being Lennie's guardian. Furthermore, the two men have a common goal. Their dream is to buy ten acres of land and gain their independence. This dream becomes the foundation on which their relationship rests and gives them the confidence to resist every attempt to break them apart—until the unfortunate and tragic end.