"To goad" is to poke an animal or person with a goad (a sharp-pointed stick used for driving cattle and similar livestock) or a similar instrument. The word is more often than not used figuratively. For example, you could "goad" someone into doing something. This may imply annoying or provoking someone into action, much in the same way an actual goad would (somewhat painfully) incite an animal to move. This mental association can help you get a better feel for the word and its more negative nuance.
"Prod" is a common word meaning "to poke" (the word is also a noun, as in "she gave him a prod in the ribs"). You would be more likely to see and hear "prod" used in its literal sense than "goad". To "prod" is also more generic. You can prod at something or prod someone with an object or with parts of your body (finger, foot, elbow, toe etc.). As is the case with "goad", you can also "prod" someone into action or "prod" someone into doing something by encouraging or persuading them.