In particular, there is nothing in Foundation that depends in any significant way on The End of Eternity.
Do you mean Foundation in particular or the series as a whole? Cause the later books in the series, particularly Foundation & Earth show strong ties to the Robot series whereas his early ones don't. I'd honestly probably avoid Foundation & Earth. I loved the story arc he was going for, but I found Foundation & Earth ultimately disappointing after the huge build up as it seemed to exist solely to tie the series together with his other works. The Hermaphrodite character also weirded me out in the way it was written. But I actually really liked his prequels.
That said, I love Asimov. His simple prose is what made it so easy to get into.