Thankfully both your Galaxy S3 and LG G4 have microSD card slots so if you don't already have a card, get one for your S3 and you can use that to copy as much of your personal files to that.
Unfortunately there's no real easy, seamless way to transfer
everything from an old Android phone to a new one, especially when they're two different brands. (iPhones and Apple's iCloud/iTunes backup and restore have us beat in this aspect.) But on your S3, be sure to go into the Settings >> Accounts and tap on 'Backup and reset' and check your settings there to be sure you've got things backing up to your online Google account. Note it would be nice if Google had made things more clear about this -- the data relative to these options only apply to Google services, so things like your personal data and files are still your responsibility.
While you're in Settings >> Accounts go into your different services (some of your third party apps should be also listed here) and be sure the check boxes and such are checked on so things tied to those specific services are getting backed up. Pay special attention to the 'Google' entry -- when you tap on it your primary gmail address should be listed and when you tap on that, a page should come up with a long list of different Google services. Be sure all the services you want saved are enabled, and if not put a checkmark in them. Then hit the 'Sync now' button at the bottom.
Your email, calendar, and contacts are most likely tied (hopefully) to your primary email account and will just transfer over from your S3 to your G4 once you set up your email account. If not you'll need to back them up as needed. The Contacts app on your S3 has an Export feature if you need it.
For your SMS texting, you might want to install this 'SMS Backup & Restore' app and then save a backup of your text messages to your microSD card. Then on your G4 you can just install that same app and use it to restore your texts.
https://play.google.com/store/apps/details?id=com.riteshsahu.SMSBackupRestore&hl=en
If you're using a third-party texting app, a lot of them include the functionality of backing up so you can just use that instead.
For things like photos and videos, you can copy those to your S3's microSD card. Or another option is to use the Google Photos app.
https://play.google.com/store/apps/details?id=com.google.android.apps.photos&hl=en
If you don't have any kind of backup solution for your photos and videos, you should think about Google Photos anyway. It's an acceptable substitute for the Gallery app but its shining feature is it will automatically back up and sync your photo library to your online Google account. If you have a pretty large number of photos and videos note it will probably take some time to upload your entire library but once done you'll have a complete duplicate of your collection. It's also accessible by going to
https://photos.google.com on any computer (once you log into your account) so you have remote access to your image archive even when your phone isn't handy. Later on your G4, you just need to install the Google Photos app and it will then sync your G4 to your library.
On your S3, for all the stuff you copy to your microSD card it's really, really important to at least double-check the things you've put on it are actually there. (Sometimes things mistakenly get copied to the wrong place, we've all done it.) In your App menu, Samsung has included a basic file manager app named 'My Files' and you can use that to check your cards contents.