Hi,Hannah
We soldem use the expression"给过去" but "递过去".
As a matter of fact,there is no great difference between "递"and"给" if you regard them as verbs.What's more,there exists a phrase "递给" which has the similar meaning of "递" and "给".But "给"is used more often.You can remember some regular collocations suah as "递文件" and "给你".
Addtionally,"给" can be a preposition which means to.For example,"给/致爱丽丝"equels to To Alice.
If my answer make you confused,please don't hesitate to ask me.I will be there to help you.