Ich soll das Buch lesen
means "I'm supposed to read this book", like when a teacher assigns you the book to read as your homework.
Ich sollte das Buch lesen.
can mean both "I should read this book", as in "all things considered, I really should...", and "I was supposed to read this book", as in "What was my homework for today? Oh, right, I was supposed to..."
The form "ich solle" is subjunctive, which means you basically only need it to make reported speech sound formal:
"Hast Du Peter nach den Deutsch-Hausaufgaben gefragt?" --
"Ja, er hat gesagt, ich solle das Buch lesen."
"Did you ask Peter about your German homework?" -- "Yes, he said I was supposed to read this book."
In everyday speech, however,
"Hast Du Peter nach den Deutsch-Hausaufgaben gefragt?" --
"Ja, er hat gesagt, ich soll das Buch lesen."
is just fine.