Often, the longer and more unusual a word, the easier it is to trace its development. It can be broken down into constituent parts, simpler roots that can then be followed back. But what about common words that form the fundamental bedrock of English?
Be is, as you might expect, a highly unusual verb. In fact, it's not really one verb at all. The modern English paradigm of be is the result of complex interactions between the forms of three different verbs which, in Indo-European, the ancestor of the family of languages to which English belongs, had the stems es-, wes- (remain, continue to be) and beu- (become). From the first we get the modern forms is, am and are, from the second was and were and from the third be and been, although exactly how that happened is a very complicated process. (For the sake of simplicity, I've omitted some alternative forms from what follows.)
The original stems worked in various ways in Sanskrit, Greek, Latin and the Germanic languages until we get to the time of Old English (up to around the middle of the 12th century AD). What were two different verbs in Gothic (am and was) have by now become one verb, am-was, the present tense expressed by the forms of am and the past by the forms of was. Be is still a separate verb and since it means 'become', it often works as the future of am-was.
The first half of the 13th century sees two major changes. In Old English, am had two plural present forms: sind(on) and aron. In southern England, sind(on) died out and was replaced by the forms of be, which then also tended to replace the singular forms. Even today in southern dialect speech you might hear forms such as 'he be' or 'they be'. Aron survived in the north and gradually spread south until (in the form are) it became standard at the start of the 16th century.
The other change to take place was that the infinitive, participle, imperative and present subjunctive forms of am-was became obsolete and were replaced by forms of be, giving us the verb in a more or less recognisable form. In Middle English, the standard past participle until into the 15th century was be until the northern form ben(e) became generally accepted.