To use your iPhone to connect to the internet via your mobile phone SIM card, you need to have a subscription to a cellular data service. This is provided by your mobile phone operator. However, in order for it to work correctly, some settings need to be configured on your phone. Very often, these settings are automatically configured on your SIM card so that they match the mobile operator that you're using. The main settings that are relevant are what's known as the Access Point Name (APN), the username and the password.
Without these settings correctly configured, it makes no difference if you have a subscription for data or not. You won't be able to access the internet through the SIM card. To check that these settings are correct, or to change them as required, you would normally access the Cellular Data section which you'll find in Settings > General > Network > Cellular on older versions of the iOS (generally pre-iOS6) and on later versions you'll usually find these located in Settings > General > Cellular. Once there, you should see the above settings and can modify them as required.
Generally speaking, the mobile operator will set these values for you and as a result, they often hide this menu option so that you can't even get in there to change the settings. This is where most people run into trouble. If your internet works through the SIM card, you'll probably never even go looking for these settings. If it doesn't, there's a fair chance that you won't be able to access the menu item to change them because your mobile operator has hidden that option. This is not related to the version of iOS that you have installed. It is directly related to the mobile operator that you use.
However - there is some good news. I found myself in this very situation with a Spanish mobile operator. In fact, it generally becomes a problem when you are using a sub-operator of one of the main operators. So, for example, in Spain you have main operators such as Movistar, Vodafone and Orange. Then you have many, many cheaper sub-operators such as Happy Movil, Mas Movil, Lebara and so on. These networks actually use the infrastructure of one of the main networks - they don't have their own physical networks. So, in my case, I was using Mas Movil. Since Mas Movil is a sub-operator of Orange, the SIM card I received had the APN, username and password configured for Orange and it had the option to change these settings hidden. So no matter how many times Mas Movil told me the settings I needed (which were not the same as Orange), I had no way to actually input them. Then I found the solution and it's worked so beautifully for me for years that I thought it was worth sharing. It makes changing the APN on your iPhone a breeze!
All you've got to do is head over to unlockit.co.nz, select the country and network that you are using and the website will generate a Profile which can be automatically installed on your iPhone. As soon as it's installed and activated, you're done. The only caveat here is that you must have access to the internet on your iPhone via wifi in order to be able to access the unlockit.co.nz website, so make sure you do it before you travel to another country where you need to change the APN!
The settings that you need for the network you are using are then stored in the profile in your iPhone and automatically override the SIM card settings. Voila! You have succeeded in changing the APN on your iPhone and you can access the internet via your mobile phone SIM!
Remember, you must go to the above website using Safari or another browser on the iPhone itself, not on your computer. Otherwise you will not be able to create a profile.