别怕 is more like don't be scared while 别担心 means don't worry. 带把伞吧 is a simple suggestion given that it's going to rain. 请 + a verb is often used as an order like in 请系好安全带 (please buckle up) and 请勿吸烟 (no smoking). Between friends and family, it's much more common to use ___ + 吧.