Is Autofocus working at all?
Obvious question first. Is the lens attempting to focus at all? If you switch the lens to manual focus, move the focus to infinity (or to closest focus), then switch back to autofocus and point at something and half-press, does the lens attempt to focus? Need to know if it's "hunting", or just not moving at all.
Autofocus not working?
If the lens isn't attempting to AF, then double-check the switches on both the lens and body. Most Nikon bodies have a switch near the lens, but it appears yours is driven by the menu. Switch off the camera, unseat the lens, and remount it, and again set the lens switch to A.
Check AE/L AF/L setting
If you've done that, and the lens isn't attempting to focus at all, then one other possibility is that you've inadvertently set the AE/L button to control focus, rather than the shutter half-press. In the Setup menu, find the setting for AE/L AF/L button, and make sure it's not set to AF/ON. Just set it to AE/AF lock.
Autofocus working, but not locking focus
Assuming the lens is focusing, but just not locking down so you can fire the shutter, let's look at the other settings. There are three focus modes:
AF-S (single focus): camera will be in "focus-priority", meaning the shutter will not release until the focus is locked (you should hear a beep, if enabled, and see a green light in the viewfinder.
AF-C (continuous): camera will be in "release-priority", meaning the shutter should fire when pressed, even if focus isn't locked in.
AF-A (auto): camera will choose between the above two. If your scene is static, nothing moving, then probably the camera would be in AF-S mode.
Try AF-C (continuous) mode
Given the camera has probably chosen AF-S for you, you are probably shooting in focus-priority. I would first switch to AF-C mode. The shutter should then fire anytime you press it. In full Auto Mode, the D3100 may not let you switch to AF-C. If not, you may have to switch into something else (M, A, S or maybe the Sports scene mode)
Once in AF-C mode, press the shutter release and the shutter should fire. If not, it's possibly not the autofocus after all.
Finally, get AF-S working
Next switch to AF-S, set the "area mode" to "Single point" and aim your centre focus point at something with a nice contrasty edge. Half press and camera should focus, and then confirm with the green light and beep.
If you cannot get the AF system to lock in, make sure you are in good light, and that you are pointing at something with good contrast.