It's customary for us to choose a medieval name to use at events. However, for your first few events, don't worry too much about a name - your ordinary first name is fine. It's entirely reasonable to simply be "Joan of Ildhafn" while you take some time to look at possibilities for a medieval name, and work out which times and places you are most interested in. Unfortunately, fantasy novels (and many historical novels) are not a good source of realistic medieval names!
In addition to names, many people take on the 'persona' of someone who might have lived in medieval times. This persona can't be an actual figure from history (don't call yourself Anne Boleyn or Ethelred the Unready, they are already taken!), nor can it be one directly adopted from myth, legend, or literature (King Arthur and Sir Lancelot are out too). Instead, an SCA persona is an invented character who might have lived in the middle ages - whose name, and perhaps background and interests, is plausible for a certain time and place - but who is not a part of recorded history. Take your time in deciding on what you want to do - your interests may change in time. In the SCA, titles (such as Lady, Sir, or Duke) are awarded by royalty for achievements in various fields, so please don't give yourself a title.
You can adopt a coat of arms, but you might want to first chat with some knowledgeable person about how the SCA handles armory. (At the very least, wait until you know what other people are using locally, and avoid well-known real-world arms.) It is certainly not necessary for newcomers to adopt a persona or medieval name at their first few events, but it is fine if you wish to do so. If you have always wanted to be a 6th century Celt or soldier at Agincourt, research the culture of the time and place, make up an identity from that period, and come as that identity to your next event.
You can get an idea of the names and personas a few of the Auckland SCA group have chosen from our "people" page.