I did the Palio this past august and loved it.
I booked about this time last year and I stayed at the Albergo Bernini from Rick's books. I have stayed there before and love the family that runs it. It's in a nice.
I didn't see La Tratta and I'm not sure if they are open to public/guests.
The trials take about as long as the race. On some trials the horses and walking around the track to get used to it for a lap or 2. I asked a similar question last year and a user on here gave me the advice to go see the trails, and make sure you are at the front of the start line for one of those to experience how the horses start. You'll see that when you get there. I was glad I did that since on the race day I didn't try to get near the start finish line. In the 3 trials i did them in different spots. The start finish line, the bottom left corner if you are facing the tower and one in the bleachers not the main square. I was shocked about how many people go for the trails, the campo was practically full.
Inside the square is Free so there is best for your money. We also had 2 people in our group get a balcony and it was nice to see their pics and video's. But I liked being down in the campo with all the locals waiving their flags.
If you can, also try to do a contrada dinner the night before. A nice neighborhood celebration with endless food and wine. We happened to do one of the larger contrada the Caterpillar for dinner.
The morning of the Palio we saw the blessing of the horse of the Ram district and the procession from the district to the Duomo. All of the contrada do the same and at the duomo is a nice parade and flag throwing. After that we made our way into the Il Campo and got a nice spot with a good vantage point. There is a parade around the track for a while before the start of the race. It was awesome to see and how all the locals waive their scarfs as the Palio banner past their section. My favorite was since the start order is not known until just prior to the start, a guy reads off the starting order and it's so quiet you can hear a pin drop on the square.
Afterwards the winners parade around ending up at the Duomo. We left the square saw some of the procession on the street just behind the square and then made our way on some back streets to beat them to the Duomo. The passion the Sienese have is a sight to see. Make your way into the Duomo and get close to the altar and wait for the winners to bring the Palio inside. Amazing!!!!
Feel free to message me if you have more questions. I was in Siena for 5 days in August for the race and I will go back for it with some more people again.