그리고 means "and" and is a conjunction used at the beginning of a sentence. 그래서 means something like "so" or "therefore," and is another conjunction that you use at the beginning of a sentence. 와 and 과 both mean "and." They are used to link two or more nouns in a list. 와 is used if the preceding word ended in a vowel sound, while 과 is used if the preceding word ended in a consonant sound. For example, "너와 나" (You and I) or "볼펜과 연필" (pen and pencil). 하고 is also "and" and could be used in the same way (볼펜하고 연필), though 하고 is more often used in conversation rather than in writing. 와 and 과 appear often in writing. However, they are pretty much interchangeable. 하고 is also used to combine actions that end with 하다. For example, 나는 운동하고 자요 (I exercise and sleep). A nuance here is that the first action precedes the second in time order, but not always. It's contextual. Consequently, an action that does not end in 하다 can be joined with -고, as in, 나는 먹고 자요 (I eat and sleep). Hope that helps.

그리고, 와,과, and 하고 all means "and". 와 and 과 are commonly used in writing. 하고 is the most general form of "and". But when speaking, it is more common to say (이)랑 especially when it's attached to people. 그리고 I often see in written form as the beginning word and not as the one in the middle of the sentence. It is better to connect two statements with -고 instead of ending first sentence then using 그리고 to start the other sentence. 그래서 means "so", "therefore", "thus", "for that reason" etc. It is just like 그리고 in the beginning of the sentence but it also has the connective -서 to bind two statements into one.