早く：can mean either (1) early; or (2) fast/quickly (mostly used to describe the speed with which one performs an action). When used to mean fast, it can also be written as 速く. Ex. (1) early - 早く来てください。(Please come early) - 疲れていたので早く寝た。(I was so tired that I went to bed early.) (2)fast/quickly - 彼女はなんて早く泳ぐんでしょう。 (How fast she swims!) - 彼は速く走れる。(He can run fast.) 急いで：means "in a hurry". It comes from the verb 急ぐ, which means "to rush, be in a haste." Unlike はやく, it has an emotional nuance (you do something fast/in a hurry because of some pressing (emotional) matter). Ex. - 彼は急いで空港にいった。("He went to the airport in a hurry.") - 朝食を食べて、急いで学校に行った。("Having finished breakfast, I hurried to school.") Both はやく and 急いで can be used in orders and commands, Ex. - 急いで準備をしなさい 。("Get ready quickly!" or "Hurry up and get ready!") - はやくしなさい。（Be quick! (i.e. Do it quickly)） I hope this helps!

早く is modifying the noun, like an adverb. 急いで is the te form and also modies a noun, or you can use it in ます form and 急いでいる to be doing an action of hurrying. Whereas the te form could be "hastily ..." 急ぎ is a connecting form as well, but it firectly attaches to another verb I believe. If I had examples I could explain it better.