Wowhead has a unique 3D modelviewer which allows you to preview any piece of gear. You can view an item, or group of items, by hitting "View 3D" on item, itemset, and transmog set pages. You can preview your character's complete appearance using the Dressing Room. You can also select the default race/gender on the modelviewer under User Settings.
What appearances perfectly match each other
The Transmog Set Database shows gear that exactly matches in game, ranging from raid sets to BoEs. While not the most creative, this is a good first step to find gear.
What goes well with my unique piece of armor or weapon
The "Transmogrified with" tab on any item or transmog set page will show the top 4 matches from Battle.net, organized by important slots, for that item. For example, all transmog sets will offer weapon suggestions, while belt pages will recommend matching pants. This is great for weapons or capes, which are not part of official transmog sets.
The "Outfits" tab on any item or transmog set page will link to user-created outfits that use a particular item.
How can I create my own outfit on my character and share it
Go to the Dressing Room and there will be options to add gear for each slot, or modify your character's appearance. To preview gear coming in a future patch, visit the Legion version of the page.
When you are logged in, hitting the "Save" option on the Dressing Room page will save it to the Outfit section.
How can I browse interesting outfits for inspiration
The Outfits section is a collection of user-submitted transmog sets. You can filter them by class, race, and gender and browse them by image gallery.
How can I see what all gear for a particular slot looks like
When browsing a filtered list in the items database, hit the "Gallery" option before the long list of items. This will bring up pictures of each image, and you can click on them to open each in the modelviewer.
How can I see what gear shares an appearance, or has the same look across colors
The "Same Model As" tab on any item page will show images of other items that have an identical appearance, both of the same color or other colors.