I've got an iPhone 5 running iOS8. It can't use a sync cable, because it came broken when I bought it used in January 2015 (no returns accepted, and no resale value if it's broken, so I just kept it, but barely use it, would prefer to stop using it altogether, and only haven't yet because there are files still stuck on it).
I just spent like two hours trying to upload my photos to Google Drive and Onedrive, and I only just figured out how, because the apps I was using seem to be designed to give the user a hard time with uploading (Drive app doesn't seem to have an Upload option, Photos forces me to select files one by one instead of hitting Select All, Photos forces me to only upload ten files to Drive at a time, drive.google.com doesn't work on my iPhone when I try it, etc).
I've figured out how to back up photos (using the Onedrive app works well enough), but I still don't know how to back up my files from the Notes app, the videos from an app called Download,and the audio files from Smart Recorder and Voice Memos. These apps don't have a "share" option built in, and I don't have or know of any app that can access those apps' files (I tried Onedrive, and it only lets me to choose files from Photos).
Is there any way I haven't thought of to transfer these files to another device? I'd prefer not to resort to copying my Notes into emails, recording the playback of my audio files with another device and filming the screen of my iPhone as it plays my video files, because doing all those things would take forever.
Best Answer
If you plan on getting another iPhone
iCloud backs up everything on your iPhone—every game's data, every app's data, your health data, your messages, your phone logs, your settings, your Homekit data, your photos and videos, everything. If you get a new iPhone, when setting it up, you can choose to set it up from a backup, and that new phone will have all of the data your current phone has. The sync is as complete as backing it up to a Mac. Also—try the Google Photos (not Google Drive) app for awesome photo backups.
If you need any help backing your phone up with iCloud or iTunes, check out Apple's support article.
Note: you can very easily get a new sync cable for your iPhone from Amazon. Here's a reliable one, and it's just $6.
If you plan on switching to Android or you're not sure what your next phone will be
Save your photos with Google Photos: download the app for iOS and it's really easy to back up all of your photos, and since Google Photos comes preinstalled on many Android phones (and is available from the Play Store on all others), your photos will be right there waiting for you.
Save contacts by going to iCloud.com and exporting them all as a VCard, then importing this into your Gmail contacts; see a guide here: https://support.apple.com/kb/PH3606?locale=en_US.
Notes synced to your Gmail account are just waiting out there in your google account, but notes synced to your iCloud account must be manually exported with Copy and Paste. Here's Apple's official guide on exporting your iCloud notes:
Emails are self-explanatory, just hook up the Gmail app on your next phone.
Calendars are pretty self-explanatory too, they should all be tied to accounts in the cloud you can sign in to on an Android phone.
Your phone number is hooked up to your Sim card, but be sure to deactivate iMessage if you switch to Android or you won't get some text messages! Here's a guide on that: http://www.imore.com/how-to-disable-deactivate-imessage-iphone-ipad
Anything else on your phone can likely be exported in its own way, and more than you may realize is tied to the cloud.