How to update iOS 5.1.1

You can't update to 5.1.1 on an iPhone 3Gs, you can only update to the current version (i.e. 6.0.1) as that is the only version that is 'signed' by Apple for it.

If you are already on iOS 5+ then you should have a Settings > General > Software Update option on it for updating directly on it. If you are on iOS 4 or below then you can only update via your computer's iTunes (you can also update via your computer if you are on iOS 5+).

To update via your computer, connect the iPhone to it and copy any purchases off the phone to your computer via File > Transfer Purchases. You may also want to copy photos and any important documents off the phone as well e.g. via the file sharing section at the bottom of the device's apps tab when connected to iTunes, via wifi, email, dropbox etc - they should be included in the backup, but it's best to have a copy of them outside of the backup just in case. You can then force a backup of the phone by right-clicking the iPhone 'Device' on the left-hand side of iTunes and selecting 'Backup'.

Then start the update by selecting the iPhone on the left-hand side of iTunes, and on the Summary tab on the right-hand side clicking the Check For Updates button

Updating to iOS 5+ : http://support.apple.com/kb/HT4972

Updating to iOS 6 will remove the built-in YouTube app and the Maps app will change.