Transfer data from iOS 7 to iOS 9?

I'm planning to transfer my data from iOS 7 (iPhone 5S) to iOS 9 (iPhone 6S). Would I suffer any loss of info as some apps may not be supported on iOS 9 because they are out-dated?

No you wouldn't

Need a computer with iTunes:
1) plug in your old iPhone and let it sync. It should tell you if you want to "download apps". Say yes
2) sync old phone with iTunes.
Find phone and there should be an option that's iPhone.
Click it and go to apps. There should be an option that says "download apps". Press it and your done. Let it sync etc.
Google sync apps to iPhone from iTunes for more help

If it's just via iCloud, yes you could lose some info like game progress, some messages, etc.

The best way to upgrade it to backup all your info on iTunes from your 5s. Then plug in your iPhone 6s, if it's new it'll ask you if you want to set it up as new or from a backup, choose backup and choose the 5s backup you just made, once it's done loading all your apps, you'll have everything the same. If apps need to be updated, in the App Store you'll see your pending updates.

If it has been synced before or done setting up and you want to transfer your old info over, you need to restore it and once it's done it'll give you the option to set up from a backup.