What I know is that a common way of saying with is using gen1 跟 or he2 和 . I know 和 could be translated as "and" but it can mean also "with", right?.
Yes, you are right. "and" "with" could have a third translation in Mandarin, which is 与. The major difference among 跟 与 和 is the degree of their formal features. 与 sounds the most serious and formal and thus more often used in written Mandarin, especially formal documentation. 和 & 跟 can be both used in colloquial language,and 和 is also used in formal context.
And about gen1, I also used this expression: gen1 .... yi1qi3, for example gen1 wo3 yi1qi3 (with me). yi1 qi3 means together, right?
Yes.
he2 wo3 yi1 qi3 is correct, too.
"I would like to be with you" (我想和你在一起) <-- correct? Yes
"I'm studying Chinese with you" (你和我正在学习汉语) <-- correct? 我在和你一起学汉语 sounds more natural.
"I'm angry with you" (translate it this way if it's similar in Chinese..else just translate it in the right way..maybe like "you make me angry"?) I don't know how it is in Chinese.
我对你很生气. with can be translated into 对 in this sentence.
"With me you can learn much" (In this case, probably "with me" assumes the meaning of "thanks to me")
It is hard to translate this sentence in Mandarin, because normally we say" you can learn much from me" 你可以向我学到很多, but "Thanks to me, you can learn much" is rarely used. How to translate it depends on what you want to express.
But I agree with Hong's translation, 跟着我你可以学到很多- "Follow me, you can learn much", but it is still slightly different from the original sentence.
我能教你很多东西--I can teach you much.
"With great effort you can succeed". Here "with" is more like a verb-"devote", so the translation is 付出努力 方可成功