Ringtones are a one-time only download from the store - and depending upon what country that you are in you might not be able to redownload music and/or music video purchases. If you are in a country wheremusic and/or videos can be redownloaded (and you haven't changed countries since buying them and they are still in your country's store) then they will show in the Purchased tab in the iTunes store app on your phone.
For the ringtones, do you have have copies of them in the Tones part of your computer's iTunes library and/or on your backup of your downloads/library (and do you have you music and videos there as well) ? If you do then you can sync them to your new phone from your computer. Or do you still have your old phone with them on ? If you do then you can connect that to your computer's iTunes and do File > Devices > Transfer Purchases to copy them over to your computer so that you can sync them.
For the music that you have on your new phone, does it show in the Music app with a cloud icon against it ? If it does then that means that it's only in the cloud, not on your phone - you can turn off cloud items showing via Settngs > iTunes & App Store > Music (under the 'show all' heading) 'off'