Knowing when to go to a tourist site is also key to saving time or getting the best vantage point. This is up to you. If you don’t mind standing in line and would rather sleep in, more power to you. But know that you’ll be fighting more crowds and have less time to see the site. Just because you’re on vacation and the sun doesn’t set until 10pm doesn’t mean these sites are open that long. Know the opening and closing times for each site you plan to visit.
Knowing the times will also help you avoid scammers in some places. Bangkok is notorious for scammers waiting outside the palace telling people that the palace is closed so they can take unsuspecting tourists to areas that further away and charge more.
Also, ask the hotel or locals on what is occurring in that area. They’ll often know if something is going on in a certain area because they’ll know to avoid that area due to the traffic or road closures.
Keep in mind that if you travel during the off peak season, some sites may be closed for maintenance or may only be open during peak tourist season. Also, depending on the time of year, getting to that destination may be difficult as well. As much as I distrust the comments section of anything, checking out comments on tripadvisor will give you insight into a specific site that won’t be published on the official tourism site.
Personally, I always try to get their first. Although I try to budget my time, this allows me to spend extra time there if I find it particularly interesting without feeling rushed to see everything before it closes. That and I will always choose to avoid crowds when I can. But this is entirely up to you.