The Bayeux Tapestry is probably the most detailed visual record we have of what soldiers on both sides of the battle of Hastings used for weapons and armor. Here is a link to a website where you can look at screenshots of pretty much the whole thing: http://www.bayeuxtapestry.org.uk/bayeux27.htm

There is of course a lot of room for interpretation when trying to draw conclusions from a piece of art, these are not photographs, and artistic conventions and technical limitations mean we always have to have a little suspicion about any conclusions drawn from a single piece of artwork. There are other artistic representations of warriors from the same region and time period which have a lot of similarities to what we see in the Bayeux Tapestry, so most historians agree that it is accurate for the broad strokes. We can also supplement art with archaeological finds, but these finds can tell us little about body-armor in this period, since no complete mail shirt from Europe survives from earlier than the 14th century, and other forms of armor would have been made of organic materials not likely to survive buried in the ground.

There is some controversy about 1. The prevalence of mail armor and 2: whether the tapestry shows Norman equipment rather than Anglo-Saxon. Assuming that mail was less common than it is shown in art, and that the Anglo-Saxons were equipped much the same as their Norman counterparts, I think I can answer your question thusly:

First we have to answer whether you are a member of the fyrd or a huscarle serving in the retinue of an Anglo-Saxon noble. If you are the former, you are probably lucky to have anything more than a helmet and shield for personal protection. Mail armor would have been the more-or-less exclusive purview of the elite huscarle. There is not enough evidence of padded cloth armor (gambeson/aketons) from this period for us to say conclusively that they would have been worn independent of mail, but given their technical simplicity, their prevalence in art from later periods, and the fact that mail was probably always worn with a padded gambeson/aketon; it is possible that fyrdmen would have made use of a quilted or padded cloth/wool garment to protect the torso. Metal helmets may well have been a luxury for members of the fyrd, but were probably commonly available to the more high status members thereof.

For personal offensive weapons, heavy thrusting spears, lighter throwing javelins, long daggers/knives (called seaxes), and perhaps small hand-axes would have been among the weapons seen in the fyrd. Generally either a thrusting spear or set of javelins paired with a personal side-arm such as a seaxe would have been fairly standard. All these weapons are well attested in the archaeological record throughout the Anglo-Saxon period.

A huscarle would have been similarly equipped with the following exceptions. First, mail body armor was likely common if not universal among these warriors. Serving in the retinue of an aristocratic lord, their war gear would have reflected the status of their lord, and gifts of weapons and armor from lord to loyal warrior-servant are commonly referred to in marital literature from the Anglo-Saxon period. Secondly, huscarles would have carried swords and the so-called "dane-axe" as personal weapons in addition to spears/javelins. Personally I think combinations of either Spear/Sword or Axe/Sword would have been the norm. It is worth noting that in the Bayeux tapestry the most common hand-weapon by far is either a spear or lance.

The shield would have been the one piece of gear no warrior at Hasting could do without. Period artwork tells us this would have taken the form of the so-called "kite-shield" at the time of Hastings, although the Bayeaux Tapestry does show some Anglo-Saxon warriors using older round shields. This is where interpretation can be problematic, the presence of a handful of round shields could be indicative of a reality that the Saxons carried round shields as opposed to kite-shields in general, and the tapestry errs on portraying Norman styles. Or it could be that round shields were falling out of fashion and only a few older warriors (the Saxons shown with round shields also have conspicuous beards) were clinging to older ways. The kite-shield would have been well-suited to the infantry fighting style of the Saxons though, so who knows.