Phone won't connect to the internet

I've only had my phone a week, and it's the first time I've had one like this. So I know I'm totally new to this and it's a completely ridiculous question. Feel free to point and laugh. 

I had a problem where I lost standard connection a few days ago, but wifi was fine so I used that. It came back quite quickly but I'm not sure quite what I did. It went again yesterday and I had no signal, and couldn't use the phone to make calls either. It just had the red circle at the top. It went at work and was still off when I got home, so it can't have been just a lack of signal. Now I don’t even have the red circle.

I've tried pressing various buttons, I've downloaded an upgrade from the Samsung site and I've even done a factory reset, but it still won't connect to the network although I can make calls again. The wifi still works so I used that to download stuff again.

I'd be really grateful for any help. I don't actually understand most of it, so can you tell me how to get to wherever I need to go! 

Thanks for the reply  I checked those two things and they seem to be set correctly. The sim card is the one from my old phone so it should be okay, although I guess they all die eventually.

If it is the account does that I have to ring the customer services for them to reset the data, or can I do it myself?

Good to hear you got help. People tend to forget that sometimes only the carrier can give information. Is it possible to share the steps (maybe update your previous post) for others on O2?