"I go home and sleep."
Translation:저는 집에 가서 잠을 자요.
I know this question is already 3 months old, but since it's an interesting one, I want to answer anyways. My Korean teacher said that the 서 particle is used in connection with the sentence part before, which basically means if you do the first thing to be able to do the second, 서 is the right word to use. Otherwise, it would be 고 (as you probably already learned). Knowing this, this sentence would probably get translated as "I go home to sleep."
"저는 집에 가서 잠을 자요."
저 - polite form of "I" (casual form would be "나")
는 - topic particle - the sentence is about you, which is why you use this one.
집 - means house, but I connection with a person (저, I in this case), it means that person's home)
에 - place/time particle, in this case it's the place particle for the verb 가다, to go and marks the place WHERE you go.
가 - conjugated form of 가다, to go
서 - 아서/어서 after verbs is used to connect it to another part of the sentence. Common, easy to remember translation is "so", but it's also used, like in this case, to say what you did to be able to do the next thing. It gives the sentence the meaning of "I go home to sleep.".
줌을 자요. - conjugated form of 줌을 자다, to sleep. This is the long form though and normally you would just use 자다, which also means to sleep.