Trip to Mount Rushmore
 

My wife and I are planning to take a trip to the Salt Lake City and Mount Rushmore area this summer, we will be traveling by train, once we get to Salt Lake City would it be better to rent a car and drive to the Mount Rushmore, Devil's Tower and other places or take a tour to these places

The California Zephyr arrives very early in the morning (if on time). Take a taxi from the Amtrak station to the 24 hour Denny's for a good breakfast. Get another taxi out to the airport to rent a car (not FOX). Plan on being to the car rental place soon after they open.
The rental car gives you a lot more options and choice on what you want to see when.

Rent a car. And don't necessarily rent from the airport, check off-airport locations for possibly better rates.

I do hope you are going to Yellowstone on this trip.

If what you want to see is Mt. Rushmore and Devils Tower, take the Empire Builder to Williston ND. Take US 85 south to Spearfish SD which is about halfway between your 2 destinations.
The westbound Empire Builder is scheduled in Williston at 11:05AM which is a whole lot better than arriving in SLC at 11:05PM (or later). The westbound Empire Builder leaves Chicago Union Station 15 minutes after the California Zephyr 2:15 vs 2PM.
I have ridden both the EB and the CZ both ways. Arriving in SLC late night and waiting for a rental car place to open is no fun.
The TRAX lightrail station is very close to the Amtrak station in Salt Lake City but their trains don't run until 7AM.

Ummm... I am pretty sure Mt Rushmore is not in Florida, wrong tagging.

Juan F. Perez please check a map and get back to us.
Do you live in Florida?

It's tagged in South Dakota for me, I don't see Florida anywhere.