It's very important to realize that slavery in ancient Greece and Rome was very different from what we normally think of as slavery. It wasn't uncommon the slaves and the slave owners have the same culture, language, and ethnicity. Slavery also wasn't necessarily forever, as there was a process called "manumission" in Rome that was a ceremony that represented an ending of slavery by either slave paying off of their debt, providing some great service to their (now former) owner, or some other purpose. It was common enough that Emperor Augustus prohibited the process for slaves under the age of 30 and limited the number of slaves that could be freed to 100 by large households.
It's also important to note that as a matter of state and large scale production slavery was indispensable. Without the labor saving devices we have now accomplishing virtually anything agricultural productive enough to trade required many more man-hours than a single family could produce. There was simply too much that needed doing by hand and slavery was one of only a couple of feasible answers.
There were a couple of different groups that opposed the practices that surrounded slavery for different reasons. The Stoics argued that all humans were manifestations of the same spirit, and therefore slavery was an external problem that interfered with the achievement of moral and intellectual detachment on both the part of slave and the slave owner. Cynics also argued that both slaves and slave owners were dragged into a maelstrom of emotional behavior by the institution. That being said, neither group mounted much of a political push and there is little evidence that they organized on the issue of slavery in particular. Christians were another group that organized against slavery, but were more interested in ending abuses against slaves than ending the institution altogether. According to tradition the Popes Clement I (92–99), Pius I (158–167), and Callixtus I (217–222) were freed slaves, but there's no evidence that they attempted to end slavery, even if slaves were given equal access and standing during religious service.
It's arguably most telling that former slaves didn't organize against slavery. The libertini or freed slaves had limited political rights (the ability to vote but not to hold office) and little other difference from that of middle-class citizens. Wealthy freedmen were common enough to warrant a caricature in the form of Trimalchio in the 1st Century fictional work Satyricon. Several prominent known or presumed freedmen include the poet Horace, Emperor Claudius' trusted friend Tiberius Claudius Narcissus, and the Brothers who owned the House of the Vettii in Pompeii.
In short, there were several groups acting as activists on behalf of slaves, but we don't have evidence of any taking action against slavery itself.
Citations:
Henrik Mouritsen, The Freedman in the Roman World (Cambridge University Press, 2011)
Fergus Millar, The Crowd in Rome in the Late Republic (University of Michigan, 1998, 2002)