The forested hills that form Bogota's eastern backdrop are part of the Reserva Forestal Protectora Bosque Oriental, a protected forest that runs for 50 km along the edge of the city. From almost anywhere in Bogota you can see them; from the hills themselves, you can see the whole city spread across the savanna below.
Several hiking trails cut into the reserve from the eastern barrios. The route up to Patios and the Cascada de la Chorrera is popular with locals. The road to Monserrate is paved and accessible by cable car or funicular, but more strenuous hikes go further north into the reserve. Bring water, sunscreen, and warm layers — the temperature drops quickly at altitude and afternoon clouds often roll in. Check with locals about current trail conditions and safety.