As well as Dropbox, you could also use Google Drive, which you may find gives you more initial space. Both are cloud storage and will be quite slow and use data (so do it on wifi!). They're good options for automatic backups, though - you can set either or both to upload photos by themselves.
You could also connect each phone by cable to a computer - if you use Windows and are asked how you want to connect, select external storage so that you can browse the device as another disc drive. Leave the sd card in the old phone and copy the folders you want to the computer. Now connect the new one in the same way and simply copy across. It's a bit more fiddly than automating it, but gives you more control.
It's also worth looking in the main settings menu for the phone (I think it's under Backup and reset) as there may be a Get Content From Another Phone option. I've used that for call logs and texts, but there were more options as well.
