After MONTHS of having this issue I finally got it fixed. I'm not exactly sure how I did it but I'll just repeat what I did in case it helps anyone else.
Ever since I upgraded to iOS 8.2 (and now on 8.3) I've had this issue on my iPhone 6 and iPad mini. Safari won't work at all. I can open the app but it just hangs when I try to load a page, and if you type something into the search/address bar it won't respond when you hit the "Go" button. I tried everything that was suggested... hardware reset, clearing browsing data and cache, etc.. It didn't help.
Two other points. I broke my iPad mini and used Apple Care + to get a replacement (thankfully). Once I restored data to it though Safari was useless again on a new device. Then, I purchased a new iPad Air 2 for work. I started with a clean slate and safari worked at first... then after I got everything set up and logged into iCloud it stopped working. That got me thinking this was an iCloud issue and it had something to do with my account and not the hardware or the version of iOS I was running.
So anyways, here is what I did to fix it.
I logged out of iCloud on the device, then cleared the cache in Safari, and then logged back into iCloud. This fixed the issue on one of the devices (the new iPad Air 2) but not the other devices.
For the two other devices I logged out and back into iCloud a few times and it didn't help. Finally I started thinking, maybe I could reset my safari settings in iCloud.com. I logged in and looked at the settings page but there was nothing there I could see to reset it. For some odd reason though as soon as I Logged into iCloud.com Safari started working on my iPhone 6 and iPad mini. Like I said, not a clear list of instructions but it's working for me now. I think just logging into iCloud.com somehow reset something and cleared the issue.
So, TRY THIS.
Go to iCloud.com, log in, and look at your settings page. Then just go back to your device and see if it's working. If not try logging out of iCloud on your actual device.