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For free, you can't miss Temple Square.
(photo from here)
The grounds themselves are beautiful just to walk around or for a photo-op (including views of the Salt Lake Temple itself as well as reflecting pools and countless gardens) but there are a number of attractions to visit and/or tour, too. Some of the highlights:
(view from Ensign Peak, photo from here)
Ensign Peak has a great view and is a short and easy hike just minutes from downtown Salt Lake City.
This is the Place Heritage Park consists of a monument to the Mormon pioneers' arrival in the Salt Lake Valley, as well as a visitor's center/shop and a village of preserved pioneer homes. There is a small admission fee.