There is a lot of confusion around these terms, so I've put together a very brief introduction to help you understand what each of these mean and how they fit together.
Let's start at the top - paganism. The dictionary definition of paganism is "a religion other than one of the main world religions, specifically a non-Christian or pre-Christian religion." The word comes from the Latin pāgānus meaning rural or rustic and was used by early Christians as a derogatory term for people in the Roman Empire who practiced polytheism (worshipped the Roman pantheon of gods). These people were also referred to as heathens and were seen as inferior to the Christians. Throughout history these terms have been applied to the practitioners of many unfamiliar religions, which are often seen to follow "false gods". In modern times, the term paganism has shed much of its disparaging connotation and simply covers religions and practices that are apart from the norm, especially those which are nature-based. The terms neo-pagan or modern pagan are often used to differentiate newer practices (such as New Age and Wicca) from the ancient pagan religions.
Next we have witchcraft. Witchcraft is also an umbrella term for religious practices involving magic and an affinity with nature, usually within a pagan tradition. (We say usually since there are also Christian, Jewish and atheist witches etc whose practices would not be considered pagan). Forms of witchcraft exist across the globe; examples include the Italian Strega, Romani "Gypsies", Haitian Vodou, the witch doctors in Africa and medicine men in Native America. Practices of witchcraft are sadly often demonised, most well know being the witch trials in both Europe and America where those accused were considered possessed by the devil and were burned at the stake or hung. Thankfully this perception has shifted and while there are still those who denounce these practices, it is generally well accepted and acknowledged by the wider community.
Modern paganism and witchcraft practices are incredibly extensive, and ever-growing. Some people have been drawn to revive and reconstruct ancient religions (including but not limited to Greek, Roman, Celtic and Norse) to worship these Old Gods and incorporate the traditions, rituals and practices into their lives. Others take a more eclectic approach and draw inspiration from many sources which is where Wicca comes in. Wicca was born in the 1950's by Gerald Gardener who was influenced by western witchcraft traditions, occult practices in Asia and esoteric literature such as the work of Aleister Crowley. He published the book Witchcraft Today in 1954 and founded a coven which developed into the practice we now call Wicca. While there is still a lot of diversity among practitioners of Wicca (such as the worship of different deities) there is a core ethical code which can be found in the Wiccan Rede which states "an it harm none, do as ye will" and the Rule of Three which holds that whatever you put out into the world, good or bad, will come back to you threefold. This sets it apart from many other witchcraft practices which work with curses or bindings for defence through to punishment depending on the particular practice.
To sum it up, Wicca is a specific type of witchcraft, and witchcraft is (usually) a form of paganism. If you are interested in witchcraft, do not think you have to be Wiccan, there are many other traditions and practices that you can follow so do plenty of research before you settle on a path! Personally, I am inspired and influenced by the ancient Celtic and Norse traditions. I am drawn to their folklore and the way they explained the unknowns of the world around them. I enjoy observing the seasons and the natural world. I like being self-sufficient, knowing how to take care of myself and I find magic in making something from scratch - such as cooking, sewing or building. I'm not one for lavish ritual or intricate spells, my witchcraft is simple and earthy. Yours will be your own.