iphone 5 slow on internet

I have an iphone 5. I have noticed that since we moved into our new apartment, the internet connection is slow. Opening email is painful. Trying to download ap updates in the ap store is almost impossible. It's crazy! It was never like this in our old place.

I know the obvious answer is to check the wifi connection. We have a secure connection that is password protected. My phone has a full wifi signal, and our computers work great. I think its the fact that we have awful, AWFUL cell service in this place. I am lucky if I get 2 bars. So I am guessing my problem is the cell signal being crappy? If so, how come the phone is using data to get online instead of the wifi signal? Is there a setting I can change to make it pull from the wifi signal first and then go to data if there isn't a wifi signal?

Or is this problem something else entirely? I can't tell you how many times I've actually had to go outside of my apartment, where there is no wifi, but a stronger cell signal, in order to update my aps. Very annoying.

Thanks.