My guide for a day in Quito will start with a nice breakfast and here there will be lots of options. One of my favorites is Community Hostel located in Old Town, for a few bucks you will get coffee or tea, a small bowl of fruits, and the options of the day varying on sweet or salty breakfast (French toast, cheese omelet, and some other options depending on the day). Or you can go to the Central Market and get tortillas de verde, fried egg, and a cup of coffee for USD 1.50, after breakfast 9am the first visit to Basilica del Voto Nacional, where for 2 USD you get access to the towers of the church (please go to the front, left tower), half the way up is in normal steps then you get to a big balcony with a great view of the Old Town, and the Virgen of Quito. After you can follow behind the balcony a wooden bridge and after that more steep steps to get to the top of the Condor Tower and have more amazing views and also get rid of your fear of heights.
After getting all this adrenaline form that visit, you should get some culture and history with a Free Walking Tour of Old Town, 3 hours of information about how the country got out of the rule of the Spanish Crown, started as a Republic, with the small time they were part of the dream of Simon Bolivar, all the calamities Presidents went through in the Republican times while you are passing through the streets all these events took place, Churches and houses with different styles from examples of the Quito School of Art (La Compañía de Jesús) to more Republican constructions (Central Bank of Ecuador, now Numismatic Museum). If you are lucky, and your guide has enough time from all the talking, a chocolate museum. And even finish with a traditional lunch in one of the many options your guide can recommend around the area.
The afternoon can have many options: a visit to the Middle of the World, to have some pictures on the Equator Line; another Free Walking Tour in La Floresta, the Hipster Neighborhood of Quito with tons of street art, nice cafes, and great views and a different arquitecture than Old Town; El Teleférico, a cable car getting you up to 4100 meters above sea level to overlook the city. According to your interest you can choose from these options, or make sure you stay an extra day and you can get to do all things Quito gets to offer.
That will give you still some time in the night to go out and share some good vibes with the new friends you might have done in your busy day adventure, if you are not too tired after all those activities and hanging out in the second highest capital of the World.
That would be my advise on the city for a day, but you can easily stay more days and get to do more around the Capital of Ecuador.