itunes

When you go to https://apple.com/itunes you say you cannot download it? What happens when you try? You need Windows 7 or later o5 almost any Mac OS version released in the past 10 years on your computer.

For your photos go to Settings/{your name}/iCloud and turn on Photos. This will sync all of your photos to iCloud (this is different from backing up). Do the same thing on the new phone and your photos will be restored to your new phone. You can also view them at https://icloud.com by logging in and clicking on Photos, or by installing iCloud for Windows on a Windows computer. On a Mac just go to System Preferences/iCloud and turn on Photos.

Here’s general information on transferring from one phone to another—>Transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support

You can backup the iPhone to iCloud, but it is much better to import your photos to the computer for safe keeping. You would also need iTunes to download the latest drivers for the iPhone to the computer. Why are you unable to download iTunes? What happens when you try?