Apart from the poetic phrases, the kind of English Shakespeare wrote in, Early Modern English, is not actually particularly hard to grasp the fundamentals of. However, I will say that if you don't fully understand the language, don't use it! It's so utterly infuriating to see people use nonsensical words and phrases like "ye pubbe" or a horrendous overuse of the word "thou". No-one will take your story less seriously if you have Tudor characters speaking in modern English, any less than they would if you wrote a story set in Spain in English -- but they will take it less seriously if you mangle the language of the day.
However, if you do want to learn the basics to try and give the right flavour, I suggest you start by reading the Wikipedia article (Early Modern English) on the subject. Also bear in mind that Shakespeare himself had an enormous impact on the development of the English language, introducing a huge number of words.