When I connect the iPhone to iTunes there is only two options

If this is your own computer:

Start iTunes. Make sure it is up to date.

Choose Store => Sign In (sign in using your Apple ID and password).

Choose Store => Authorize this computer (you will be shown your Apple ID, enter your password).

Connect your iPhone 4.

Choose "set up as new iPhone" if you are asked this question.

Click on the iPhone icon on top.

Your iPhone 4 details will appear: capacity, serial number etc.

You will see: "Update Software" (if update is available) and "Restore Software." Skip those, scroll down.

Important Note: if your iPhone 4 has an iCloud account on it and you know the user name and password for sure, proceed. If not, stop here and let us know for further guidance.

Tick/check "Back Up to This Computer"

Then click on "Back Up Now"

You may be asked to "Transer Apps / Purchases." Choose yes.

Back up will start. Once it is over, click on "Back Up To This Computer" again.

It should take less time now.

Once done, unplug your iPhone 4.

Note: if your iPhone 6 has not been turned before: insert a SIM card. Start your phone. Set it up as new. Skip to the main screen without logging in with your iCloud account, without adding a passcode, or anything of the like.

Once done, connect to the computer.

If you are not asked to, click on "Restore Back Up" (CAREFUL: NOT "Restore Software).

You will see a list of available back-ups. Choose the one that you had just made.

Wait for the iPhone to complete the restore. It will restart. Don't unplug it. After it restarts, it will also transfer your Apps. Keep your iPhone connected until all the Apps are transferred onto your new phone.

Hope this helps.