Quito is a very beautiful, well-preserved and tourist-friendly city. You can safely walk around the city center and become amazed at its beauty, and visit a lot of Baroque buildings, especially churches.
While in Quito I visited some of them. The ones I would definitely recommend are Compañía de Jesús and San Francisco. The first has an admission ticket and you can join a guided tour inside with no extra cost. You can visit the second for free, both the church and the cloister, but the museum it hosts has an entrance fee. You can also take a free tour guide there. Both are great examples of the Escuela Quiteña, the buildings themselves and the works of art you can admire inside.
Apart from those two, I visited the Cathedral, the Monasterio del Carmen Alto and the Museo de la Ciudad. They are all interesting examples of the same artistic school. The Museo de la Ciudad also hosts a very informative exhibition about Quito's history that I strongly recommend.
Outside the city center, I recommend visiting the recently refurbished Museo Nacional, where you can learn about Ecuador and its culture. And a bit further away, the TeleferiQo, where you can have a full view of the city from 4000 metres above sea level, is a must. You can see the highest volcanoes of Ecuador (Chimborazo, Cotopaxi, Cayambe...) from there too.
I happened to be in Quito during the 40th anniversary of it becoming a World Heritage Site. On the 8th September, there was a church bell concert in the city center. You could hear the bells of the main churches and convents in Quito chime at the same time in a coordinate way. It was an intense moment.