Laguna Chillata is a popular hike from Sorata. It's a pretty spot with great views of the surrounding sierra and Lake Titicaca. The former trail is now mostly a road, so you can take a taxi to La Mina and hike one hour up to the lagoon. On the right-hand side of the lagoon are ruins of a small Incan stone village. It's worthwhile taking a guide as it's easy to get lost.
A steep ascent the following day can take you up to Laguna Glacial, a top spot where you can watch big chunks of ice cracking off into the water. It’s at 5100m, so take it easy; the altitude can make it a tough climb.
It's possible to overnight, and leave the tent and your gear at Laguna Chillata. However, it’ll get nicked if you haven’t brought a guide, who can detail someone to watch over it.