Right off the bat, cancel the hotel reservation and get a room in Old Town. The hotels in Old Town aren't as nice, but you'll be much better off. Seriously. Old Town is where you want to be, even if it means abondoning the other wedding guests.
Try to wake up a bit early. It's always sunny like San Diego in the morning and it gets cloudy in the afternoon. Take pictures early in the day.
Spend your first day walking around the city. Take in the markets. Have a cold beer on the steps of the Plaza de Francisco. The old misson there is the oldest building in Quito. Hike over to the Virgin of Quito statue for a panoramic view of the city. Then make your way to the new cathedral they've been building since the 1950's. It's still under construction technically, but it's hard to tell. You can take a self-guided tour and climb all the way to the top of the bell tower and ring the bell. They would never let you do this stuff in America.
There's a store or cafe on every block that will sell you a beer for almost nothing. Take advange of this opportunity often.
Quito is good launching point for other adventures.
If you are reasonably fit, hire a guide and climb to the top Illiniza Norte. You absolutley WILL NOT regret it. It takes two days though. You spend one night on the mountian. A good jacket and some hiking boots are all the gear you'll need.
Take a train trip through the mountains. You can climb out of the train car and sit on the roof to get a good view. When the train stops along the way, locals will sell you plates of food trackside.
Lots of other day trip options into the mountains as well, depending on what you are into.
One night I came across a restaurant called the Texas Ranch. They had a UT flag on the wall. I showed the owner my UT tattoo and two free drinks showed up at our table.
I have to warn you about the food though. Don't expect much. It's a poor country and fine dining is not high on their list of priorities. Don't go out of your way to find good grub. It you do find it, you're missing what Ecuador is all about. Just be happy with a plate of chicken and rice in a cafe for $3. Savor all the Club and Pilsener beer you can get though. Good stuff.
The US dollar is the official currency. Bring a lot of small bills.
We didn't catch the train to/from Quito so I'm not sure where the train station is located. I don't recall it being near Old Town though. There's a streetcar system near Old Town that can get you around town. I rode it somewhere one day, but generally I'm not one to waste precious vacation time waiting on public transit when a taxi is available at a relatively low price.
Read more about scenic train rides here, including at least one trip that leaves from Quito. Link