Helium for apps and data (uncheck the "data only" box if you want it to back up the apps). I use
App Backup & Restore to back up my apps, because it makes apk files, so I can install one app if I want - from My Files even, you don't need the backup app to restore an apk file. (Helium will tell you to download a file to your PC if the phone isn't rooted - that's legitimate. It's looking for something, it can't get it from an unrooted phone, so the file kind of fakes it.)
I use
MyPhoneExplorer to back up just about everything else, texts mainly. You can use USB, wifi or Bluetooth to connect, but since you're probably already using USB for other things, set it to use USB and it'll find the phone and connect to it. (IIRC, you have to name the phone when it first finds it, from then on it just runs.)
Contacts, if they're Google contacts (they should be - not Device, Phone or SIM contacts) are backed up at
Google Contacts already, if you have contact sync enabled (it is by default). The new phone will restore from there automatically.
For contacts that aren't Google contacts, you can either just use MyPhoneExplorer to back them up or you can (and should) convert them to Google contacts. (Export to the SIM or SD card [depending on which Contacts app you have - if you can do both, just pick one, it doesn't matter], then import from the SIM or SD card, to the Google account. If you get duplicates, you can use the Merge Duplicates in More at Google Contacts.)
Any other files, just drag to the PC. If things like album structure is important, drag the entire top folder - like Music for your music - so all the albums are kept intact. (If you're copying a huge mass of files, like hundreds of songs, make sure the charger is plugged in and that you're not using a USB nub - most of them can't handle that kind of heavy-duty work. Plug directly into a port on the computer.)
Copy the carbon folder (that's Helium's folder) and the App Backup and Restore folder, if you use it, to your PC.