To piggyback on /u/cdesmoulins comment, I think reading Seven Pillars would probably be the logical place to start. Yes, it’s the basis of the self-made “Lawrence myth” but so much of the interest in him is separating the fact from fiction of that myth. Aside from anything else it’s also wonderfully written—there’s a reason for his enduring popularity.

If you really want to dive into particular minutiae, there’s also his authorized biography by Jeremy Wilson. I haven’t read it cover to cover but as such biographies often are it’s a bit of a tome with quite a lot of verbatim copies of his letters and writings and I’ve found it to be quite useful in answering a number of specific questions I’ve had in consulting it.

For web resources, you might start with Jeremy Wilson's T.E. Lawrence Studies site, which includes biographical info about Lawrence as well as reference materials and full texts of Lawrence's books and correspondence. Published biographies of Lawrence are sometimes a mixed bag, but Michael Korda's Hero: The Life and Legend of Lawrence of Arabia is a fairly recent and fairly comprehensive biography available in a number of formats. It's totally possible to go down a rabbit hole on specific T.E. Lawrence-related controversies (his political views, the political context he arose from, his sexuality and/or trauma, the truthfulness of specific aspects of his descriptions of events in Seven Pillars etc.) -- there's a lot of academic writing about the particulars of Lawrence's life, if you're interested in going on deep dives on specific topics or comparing between biographers' accounts of specific incidents. Seven Pillars Of Wisdom is out of copyright in Australia and is available online as a result; Wilson's site hosts the full text of Lawrence's The Mint here, his account of his time in the RAF.