Can I store all the content of my previous iPhone in a new iPhone using iTunes? I have an iPhone 4S with multiple backups in my computer and I'd like to buy a new iPhone.

Can I connect the new device to the computer and immediately store all the content from the previous iPhone? I want to move all the photos, messages, etc.
Yes.

The easiest way to do this is to do a full backup of your iPhone 4s via iTunes.

Then you would connect the new iPhone to iTunes as well and then do a Restore from the iPhone 4s backup to your new iPhone.

Important 

The iPhone 4s was available in various sizes, namely 8GB, 16GB, 32GB and 64GB. So you will need to ensure that your new iPhone has as a minimum the same capacity as your old one.

You should also select the "Encrypt local backup" option as this will "allow account passwords, Health and HomeKit data to be backed up". Of course, in your specific case, an iPhone 4S doesn't have health and home data.