If you want something that fits in directly with the time period of the Forgotten Realms as depicted in Phandelver then the new Princes of the Apocalypse adventure starts out just to the east of Phandelver.
I have the book and the initial adventures are in the Dessarin Valley. The north end begins at the eastern terminus of the Triboar trail as you can see in this map.
Wyvern Tor is in the Northwest corner.
The premiere website for all things Forgotten Realms is Candlekeep
You could get a copy of the original information on the Sword Coast from D&D Classics. I recommend Waterdeep and the North. I have a copy myself and it has a lot of information about the Sword Coast area. However it is for an earlier time period than what in Phandelver.
D&D Classics has Forgotten Realms material spanning all eras.
The most recent material they have on Neverwinter, is the Lost Crown of Neverwinter for 4e. I played and referee as part of a game store campaign and it quite good and has a lot of roleplaying opportunities with different factions. Although it is a bit episodic as it oriented to a weekly D&D Encounters session.
What is a really good Neverwinter resource is Jaquay's the Savage Frontier from the original Forgotten Realms run. Again it is set in a earlier time period.
You may want to look at this Candlekeep forum thread for Neverwinter resources.