"저는 식당에 가서 먹어요."
Translation:I go to a restaurant and eat.
X에 먹으러 가요 - I go somewhere to (in order to) eat.
X에 가서 먹어요 - I go somewhere and (only after arriving there) eat.
Note that in many cases, including this one, 서 can be replaced with ㄴ다음.
식당에 간 다음 먹어요. - I eat after going to the restaurant.
X에 먹고 가요 - I go somewhere and (simoultanously) eat / I go somewhere while eating.
Quite conveniently, 서 may also mean "so" is English, as it may mean not only that one action happens after another, but also that one of them is the cause of the another.
눈이 와서 산과 들이 모두 흽니다. - It snowed, so the mountains and fields are all white.
늦어서 빨리 갑시다. - It is late, so let's go quickly.
아버지는 시간이 없어서 같이 영화관에 가지 못했어요. - Dad had no time so we didn't go to the cinema together.
You have to decide by yourself on how to translate given Korean sentence. Usually, only one of the two meanings ("and then"/"so") fits better than the another.
Finally, 서 may also mean "that" and "for" (great, another bunch of seemingly unrelated meanings) when used with 고맙다/감사하다/미안하다/반갑다/좋다 and other, similar words. The usage is following:
도와줘서 고마워요 (고마와요 acc. to North Korean conjugation) - thank you for helping me.
가르쳐서 감사합니다 - thank you for teaching me.
방해해서 미안합니다 - sorry for interrupting (방해하다 - to interrupt)
오늘 날씨가 따뜻해서 좋다 - it's good that the weather is warm today.
And an example, that most of you already know very well:
만나서 반갑습니다 - I am glad that we met.
Note, however that this last meaning isn't actually that different from the second one. 도와줘서 고맙다 = Thank you for helping me = "You helped me so I thank you". 만나서 반갑다 = I am glad that we met = "We met so I am glad".