Like the Academy, the Lyceum was a public gymnasium. Aristotle studied with Plato at the Academy for twenty years, leaving Athens after his death. After some time away in Assos, Lesbos, and Macedonia, Aristotle returned to Athens and it was during this second period in the city that he started to teach at the Lyceum, with his close friend and collaborator Theophrastus. Together they studied a wide range of topics, living a life devoted to theoretical contemplation.
At some point Theophrastus bought a property nearby as a base of operations for the school, and the Lyceum continued under his leadership after Aristotle's death.
Aristotle considered contemplation to be the highest form of human activity, generating the greatest happiness.
Aristotle, Nicomachean Ethics 10.8 'But that perfect happiness is a contemplative activity will appear from the following consideration as well. We assume the gods to be above all other beings blessed and happy; but what sort of actions must we assign to them? Acts of justice? Will not the gods seem absurd if they make contracts and return deposits, and so on? Acts of a brave man, then, confronting dangers and running risks because it is noble to do so? Or liberal acts? To whom will they give? It will be strange if they are really to have money or anything of the kind. And what would their temperate acts be? Is not such praise tasteless, since they have no bad appetites? If we were to run through them all, the circumstances of action would be found trivial and unworthy of gods. Still, every one supposes that they live and therefore that they are active; we cannot suppose them to sleep like Endymion. Now if you take away from a living being action, and still more production, what is left but contemplation? Therefore the activity of God, which surpasses all others in blessedness, must be contemplative; and of human activities, therefore, that which is most akin to this must be most of the nature of happiness.
This is indicated, too, by the fact that the other animals have no share in happiness, being completely deprived of such activity. For while the whole life of the gods is blessed, and that of men too in so far as some likeness of such activity belongs to them, none of the other animals is happy, since they in no way share in contemplation. Happiness extends, then, just so far as contemplation does, and those to whom contemplation more fully belongs are more truly happy, not as a mere concomitant but in virtue of the contemplation; for this is in itself precious. Happiness, therefore, must be some form of contemplation.'
After: The Museum of Ancient Greek Technology
Close by is the Museum of Ancient Greek Technology where you can see many ancient inventions reconstructed, including Plato's alarm clock (further details).
Further Reading
The best biography of Aristotle, which has lots to say about his school at the Lyceum, is:
Carlo Natali, Aristotle: His Life and School (Princeton: Princeton University Press, 2013), esp. chs 2 and 3: