We will have 2 days in Madrid in August, and want to visit the Royal Palace and Prado. Are there certain times of day that are less busy for each of these attractions? Is it necessary to purchase speck "skip the lines" tours in order to avoid standing in long lines to enter?

What day are you going? The changing of the guard used to be Wednesday. That meant no entrance to the palace and crowds

I think with the museum the thing to avoid is the free entrance times

In October we had bought combo tickets online for Royal Palace and Galería de las Colecciones Reales. Our Royal Palace timed tickets were for 10 am (when it opened). That worked out great. For the Galeria we could enter anytime during that day (not very crowded). On a separate day, we had 1 pm entry tickets for Prado. Very crowded. Glad we had seen the Prado ten years earlier. Crowds there were a bit much. Loved the Galeria. Below is day at Royal Palace and Galeria.

https://travelswithmaitaitom.com/cha...-a-king-queen/

While not an answer to your actual question, I'll note that the time it takes to visit the Prado depends on one's interests. I spent 10 glorious hours there.

How long do you think we should plan to spend at Royal Palace?

We will arrive there on a Saturday morning and depart on Monday morning.

When I went to the Prado, I was very glad that I bought my ticket ahead of time and could just walk past the very long line and go right in! This was not with a tour. I bought the ticket on my own.
I did wait in line at the Royal Palace, don't know why I didn't buy the ticket ahead.

"How long do you think we should plan to spend at Royal Palace?"

We went through in about an hour and 20 minutes.