There are a lot of fun places to take kids in Salt Lake City. I thought I'd share a recent low-key (and low crowds) outing I took the kids on recently because it was (A) free, (B) easy, (C) flexible re: time, and (D) was a great place to take photos.
The Utah State Capitol Building is open to the public seven days a week (unless it's a holiday) and it's free to take a guided tour or simply walk around by yourself. I'd been a few times when I was younger but I hadn't taken the kids so they were really excited to see what the building looks like on the inside.
There are some historical displays in the lower level and if you visit at the right time you might see some of Utah's politicians at work, but the star of the show is the main floor with the marble dome and the murals. I was a little worried that the kids wouldn't be engaged for long once we were inside, but luckily the website has a children's activities page that included a scavenger hunt and the kids LOVED finding all the little elements.
A couple quick notes about taking kids - the website asks that if you use the worksheets or games provided on the website that you don't do them on the guided tours since that is distracting to the tour guides. I also wouldn't take markers/crayons/pens if you can avoid it because there aren't good places to sit and color. I took clipboards with sheets of round label stickers so the kids could just put stickers over the pictures as they found stuff.
Also, there's no posted rules, etc., but it is a working building with guided tours going on constantly so there's an understanding that everyone should keep their voices low. My kids do a TON of museum and library visits and are used to using their whisper voices so if that's something you aren't sure about, you might want to chat about it before you get inside. I was a little worried that the toddler would be more excited about hearing the echoes of his little voice as he ran shrieking through the Capitol but he did fine.
Also, I follow enough of our state representatives on social media to know that it isn't uncommon to arrange for a quick chat with a Senator or Representative and if that's something your family would really enjoy you might want to reach out to a few interns to see if anyone can squeeze you in.
Our kids are still too little to be able to participate in any meaningful/memorable conversation about government, at least as far as I can tell, so we were strictly scavenger hunt, a quick self-guided tour, and then off to the gift shop across the street.
Last tip - the gift shop is TINY and so expensive but it has all the "from Utah" stuff you'd expect. If you have any wiggle room, though, the best Utah souvenirs are found at the gift shops at This is the Place Monument (historical Utah), The Natural History Museum of Utah (natural Utah), Red Butte Gardens (artisan Utah), or "Everything Utah" downtown by the temple (everything else Utah but a bigger selection and better prices).