"Precede," according to Merriam Webster's dictionary, means "to go before." For example, in one film adaptation, the character Sherlock Holmes says to Professor Moriarty, "I prefer that you precede me at all times," asking the criminal mastermind to walk before him to avoid a possible attack. "Proceed," according to Merriam Webster, means to continue on. For instance, the words "Proceed to checkout" is found on Amazon.com and many other retail websites.
You can also use "precede" to mean going back to a previous history or idea, such as "His reputation precedes him." Thomas Aquinas suggested that "precede" gives some things priority over others, as in "Love must precede hatred." The word "precede" also forms the word "precedent," meaning a prior behavior or an important event. You could say, "There is a legal precedent for this case."
"Proceed" is much more straightforward, as it means to go ahead or to continue something you've already started. "Proceed to demonstrate your theory, professor," the eager student might ask. The engineer who is ready to continue might say, "Our work must proceed." Oxford Dictionaries notes, however, that the word "proceed" can also mean "originate," as in the phrase, "all power proceeds from God."