Hi. Welcome to the Ubuntu Forums!
I am not using Linux Mint currently so can't say if Shotwell is present there by default or not. It is present in Ubuntu's official variants though.
For installing software in Ubuntu and its derivatives, you should preferably use the Software Manager included in the installation which in Ubuntu's case is Ubuntu Software Center. For Linux Mint, I guess it is Software Manager. Again, I can't be sure as I have never used Linux Mint.
Anyhow, the global method across all Debian based Linux distros is to use "apt-get" for installing software. To use it you will need to open up a "Terminal" window and run this command:
It will ask for your password. Just type your password and hit <Enter> as the Terminal won't return any feedback as you type the password.
sudo apt-get install shotwell
Once installed, the software will appear in your application menu which you can drag and drop to your desktop for easy access.
You should be using the repositories (both apt-get and software manager download from the repos) for installing new software instead of downloading the software from developer websites. This way, you won't need to install it manually after downloading and you will always be updated to new versions automatically.