A Mormon Perspective
I am a Mormon. If you are not a Mormon and are interested in visiting Temple Square, here are some thoughts about how to enjoy it. First, you should decide if you want to see the temple only or if you want to also experience a bit of the religion. In my opinion, there is no better way to have an anonymous brush with the Mormon Church than at Temple Square. If you only want to see the building, walk in through one of the gates, walk around, and when you are approached by missionaries, all you have to do is politely decline. One hint is that if you wear nice business attire, dress, jacket, tie etc, you will not stand out. In any case, this is the Mormon Mecca. I suggest dressing nicely out of respect, anyway, but doing so will also help you go unnoticed, if that is what you want. It's sort of like how I went to NYC once and I wore a dress shirt and tie and nobody bothered me, but my friend wore a sweat-shirt that said "Iowa" on it and was amazed how people could tell he was from out of town.
If you want to understand a little about the religion, as I said, this is the best way to do it. The temple grounds are staffed with dozens of "sister missionaries"; young volunteer women who have left friends, family, college and jobs and are spending 18 months at their own expense, in order to answer any questions you might have. They come from all over. Probably a little more than half are US citizens, from California, New York, Arkansas, etc. The remainder come from all over the world; France, Brazil, Mexico, Thailand, India. They all wear an emblem on their name tag showing where they are from. As a matter of courtesy, I suggest being respectful and polite. That's exactly what you will get from them. You find out all you want about the church and never have to give them your address and phone number. They will probably ask your name and where you are from, as a simple matter of conversation. Make up a name, if you are concerned about privacy.
For me, the highlights within Temple Square are the Temple (exterior), the Tabernacle, and the south visitors center, which houses a scale model showing the interior of the Temple, as well as a small presentation of the building of the Temple.
Years ago, the front (East) facade of the temple was behind a gate. You could look at it, but not go up to it. At that time, the only access to it was for wedding pictures. Now, the front of the building is accessible by all. You can even walk up the steps of the Temple and have a friend take your picture. As a Mormon, I was surprised and excited the first time I saw that.
The best days to visit Temple Square are Saturday and Sunday, but for different reasons.
Saturday mornings give you the best chance to see a bride and groom having their wedding pictures taken in front of the Temple (East facade), which is a memorable experience. For Mormons, marriage is the single most important moment in life. Mormons believe that they are married, not just for this life, but for the life to come as well, so observing a wedding party at the Temple is momentous. If you do see a bride and groom, take a moment to reflect that a short while before, they were inside the Temple, where they were "sealed for time and all eternity" in the religion's highest ceremony and that they have just emerged to have their pictures taken on the steps. This is as close to experiencing the Temple itself as you can get without going inside.
Sunday mornings offer you the chance to attend the free Mormon Tabernacle Choir concert in the Tabernacle. I did this for the first time about 10 years ago and found it to be very enjoyable. Note that no weddings take place on Sundays.