"赶紧去请" can be translated into English as "Hurry up and go invite (someone)" or more naturally, "Go and invite (someone) right away" or "Quickly go and fetch/ask (someone) to come".
Here are a few examples depending on the context:
Hurry up and go invite Professor Li. （赶紧去请李教授。）
Go and fetch the doctor right away. （赶紧去请医生来。）
Quickly go and ask the manager to come here. （赶紧去请经理过来。）