Hello BColePB,
It sounds like you got a newer phone and signed into your iCloud account but do not see your iTunes store content. There are a couple of ways to get that stuff onto your phone. The faster and easier way would be to sync the music to your phone with iTunes, and this article will help you do that:
Sync your iPhone, iPad, and iPod with iTunes using USB
The other way requires you to have a backup of your iPhone 4s, which you should then be able to restore to your iPhone 5s with iTunes or iCloud. iTunes would be the faster option in that case and again this article will help you do that:
Transfer content from an iPhone, iPad, or iPod touch to a new device
You can also sign into your iTunes Store account in your settings to see your purchases and stream them from the cloud:
iTunes Store settings - iPhone
To set options for the iTunes Store, go to Settings > iTunes & App Store.
Thank you for using Apple Support Communities.
All the best,
Sterling