You have to sync your old iPhone with iTunes, then connect your iPhone 5 to iTunes and restore it. One tip: While backing up your old phone go through each tab in iTunes and make sure you select exactly what you want to keep.
I used an app called Photo Transfer to move photos wirelessly from my Iphone to my Ipad. ( Also used it to move photos from my wife's Galaxy S4 to the Ipad). Works well. Available from the App Store and Google Play store.
The safest way I would recommend is using iTunes. Just plug your iPhone 4 into your computer and click 'Back up' (( it should be on the summary page )), let it backup and later on just plug in your iPhone 5 into your computer again and click on Restore From Backup. Voila Do not forget your contacts too!