There are lots of high-quality books about the First Crusade, and the crusades in general. There are lots of more popular books that might be easier to find, but generally any popular books about the crusades are actually not very good. So hopefully this list is helpful:

Thomas Asbridge, The First Crusade: A New History (Oxford University Pres, 2004) - this is the most recent academic history of the First Crusade, and the one I usually use to answer questions here

Jonathan Riley-Smith, The First Crusade and the Idea of Crusading (University of Pennsylvania Press, 1986) - a good look at the background and the development of the concept of a crusade

Jonathan Riley-Smith, The First Crusaders, 1095-1131 (Cambridge University Press, 1997) - this one has more emphasis on the people who went on crusade, and why they went

For the crusades in general, I suggest:

Hans Mayer, The Crusades (trans. John Gillingham, Oxford University Press, 1972) - a bit old now, but still a good summary. Very heavy on names and dates though

Jonathan Riley-Smith, The Crusades: A History, 3rd ed. (Bloomsbury, 2014) - personally I find this to be a great school book...like imagine you were studying in a class about the crusades in a university, everything you'd want and would need to know would be in this book. I'm not sure if that makes it "accessible" exactly, but it's a great textbook at least

Thomas F. Madden, The New Concise History of the Crusades (Rowman and Littlefield, 2005) - accessible, but maybe a bit too concise. It's very short

Paul M. Cobb, The Race for Paradise: an Islamic History of the Crusades (Oxford University Press, 2014) - a newer book, looking at the crusades from the other side, from the Muslim perspective

Jonathan Phillips, Holy Warriors: A Modern History of the Crusades (Random House, 2010) - another newer history from the European perspective

Susanna A. Throop, The Crusades: An Epitome (Kismet Press, 2018) - if you're looking for something very concise, this is even more concise than Madden's history! But personally I think it's much more accessible/readable

Also check out the Crusades Reading List for more possibilities!