Why this order
Machu Picchu is the longest day and the hardest one to recover from if you arrive tired. Leaving it for last gives you two days in the area before the train day, and puts the city, which you can explore without rushing, at the start. If you arrive in Cusco the same day, count it as a rest day and read the altitude sickness guide.
What to book in advance
Machu Picchu is the first thing to lock in, because the ticket and the train depend on the date. Once that date is set, fit the Sacred Valley and the Cusco tour into the days before it.
What doesn't fit in 3 days
Three days leave out Humantay Lake, Rainbow Mountain, and Moray and Maras. If you can add days, see the 5-day plan.