While I'm loudly unhappy about the current state of iTunes, the unfortunately named iTunes App Store on iOS usually does just fine. Sometimes though, you still get an app you tried to buy or an update that for some reason can't seem to download successfully. If you end up with an app in limbo, here are some tips for clearing up whatever ails it and getting the download/install to finish.
First off, wait until you get an error. You think it'll be immediate, but iOS tries really really hard to do everything it can think of before it gives up, so sit tight and see if you can get a screenshot so you can look up that message later. A good way to tell if it's still trying is that little gear in the top left corner of your menu bar. If you see that rotating, try to give it more time.
Try rebooting the device and doing the same thing again. Some say this is the definition of insanity, but sometimes a reboot clears up something stuck that might be holding you up.
Next up, sign out on the iPad, by going to iOS Settings > iTunes & App Store > and tapping your Apple ID. Then restart and immediately sign back in. Like rebooting, sometimes this can shake out something that isn't quite right. Since you're signing right back in, this won't affect any of your apps or email or anything.
You'll see your Apple ID, tap that and sign out.
If none of those steps work, try signing out on your iPhone, then signing out and back in on a different device. For example: Sign out on your iPhone, then sign out on your iMac, then sign back in on your iMac, then sign in on your iPhone.
You can turn yourself about too, it's optional but couldn't hurt. If you're still stuck, try this: Sign out on your iPad. Then go to iOS Settings > General > Reset > Reset Network Settings (Note: You will have to sign in to saved Wi-Fi networks again if you do this.) Restart your iPad, then get back on the Wi-Fi. Then go to iOS Settings > iTunes & App Store and log back in.
At this point, if you still can't get App Store downloads or updates to work, you're on to the big steps: Make sure you have a recent backup and (factory) reset and restore. Failing that, you're on to (factory) reset and rebuild instead of using a backup. Should you take it in before doing these things, it's possible you'll be told to go home and try factory reset and come back, so it's best to do it before you go in.
If you have made it through all these steps and still nothing is working, make your appointment at the Apple Store. At this point, if you have done all of the above and still have no App Store joy, your device probably needs either an exorcised or replaced.
(Note: These steps can also help with music or movie downloads that are stuck; but most of the time a troublesome download is an app.)