If you visit in the summer, check the opening hours of the Registan ahead of time. There's an international dance competition held in mid-August and the opening times will vary depending on practice and performance times.
It's quite hot in July and August, if you can visit in the spring or fall it will be more enjoyable.
The Timur Mausoleum is at it's best lit up in the evening, after the crowds have died down significantly, right before closing. It's best to do this at the end of the day.
At the playground you pay by the half-hour, but they don't check in. Pay for the minimum amount of time you think you'll stay and then don't worry about it if you're there longer.