Just a quick report. After deciding that Babrius is too hard, I started working on the JACTG Reading Greek. I’m about 150 pages in. I treat the book as if it were a anthology of selections, mostly passing over the grammar instruction.
When a sentence puzzles me, as frequently happens, I write the sentence by hand, giving a separate line to each phrase. I have a small Chromebook nearby, with the Crome browser tabs open to Wicktionary and the Perseus Word Study tool for parsing the hard words. Often I can find online English translations of the passage under study which I put in another tab.
Based on the material found at the address indicated by Joel, I have made some arrangement, for my own study, collecting the various versions available in one place. I am not sure whether this might be also useful for others, however, here it is: https://vasilestancu.ro/aesop-babrius(primary).html. This time, the primary ‘key’ for ordering the fables is ‘Babrius’; I intend to complete the arrangement and order the fables by the other authors also, i.e. Aphthonius, Chambry, Syntipas.
If you’re interested in reading poetry, you might want to try elegiac poetry – Callinus and Tyrtaeus. They’re quite accessible. They’re in the Loeb Greek Elegiac Poetry along with Solon, Mimnermus and Theognis.