This novel has been suggested many times by a few people. I don't know if it falls into "classics" territory, I might be too young to judge even :)
The Sun is about to go Nova. Earth and Moon have ceased their axial rotation and present one face continuously to the sun. The bright side of Earth is covered with carnivorous forest. This is the Age of vegetables. Gren and his lady - not to mention the tummybelly men - journey to the even more terrifying Dark side. One of Aldiss' most famous and long-enduring novels, fast moving, packed with brilliant imagery.