**Changing my router settings to WPA2 solved my issue.
Situation recap:
Since purchasing my iPhone 6 64 GB AT&T on launch day, I had a very slow connection speed on my home Wi-Fi network. I was able to connect to my Wi-Fi network, however, the connection was quite poor (download speed averaged .01 - .56 Mbps). This is not my first rodeo (or iPhone for that matter) so I attempted to resolve it myself by: restoring from an iCloud backup (twice), restoring as a new device from iCloud (twice), hard reset numerous times (holding home button and power button at same time), resetting network settings, resetting all settings, etc.
I finally called Apple today and spoke with a rep who was of no help (she concluded that my slow connection was due to the Wi-Fi calling feature not being supported by AT&T until early 2015). My second conversation with a rep ran diagnostics and she stated my phone's Wi-Fi was labeled as normal or that it "passed" its test. She transferred me to a senior rep and he wanted me to restore my phone (again), but through iTunes this time. If that did not fix my issue, then visit a local Apple Store or send my phone to Apple to be tested. After ending our phone call and at the start of attempting to restore my phone via iTunes, I came across this discussion page. Changed my router's settings (Linksys router) from WEP to WPA2. It has been roughly 30 minutes since then and my Wi-Fi connection is flawless.
Thank you, guys!
P.S. the last Apple rep I spoke with gave me his office number and I called to leave a voicemail about the WPA2 solution. Hope this gets resolved for others!