If I received a Notice to Vacate my home in the mail a few days ago, how long do I have before an eviction notice is served?

There is not enough detail in your question to provide an answer: the time you have before an eviction action could be commenced will vary with the grounds for eviction. For example, if the home was foreclosed upon and the new buyer wants to live there, you would be entitled to 90 days; if you are on a month to month lease and the landlord decided that he wants you out, that's 30 days notice; if you have intentionally or grossly negligently damaged or destroyed landlord property, that's usually 3 days; nonpayment of rent typically allows a landlord to immediately try to evict you. So the reason for the alleged eviction is a critial part of answering your question.