api docs: Restructure integrations section and add an overview.

templates/zerver/api/integrations-overview.md

@@ -0,0 +1,65 @@
+# Integrations overview
+
+Integrations allow you to send data from other products into or out of
+Zulip. Zulip natively integrates with dozens of products, and with hundreds
+more through Zapier, IFTTT, and Hubot.
+
+Zulip also makes it very easy to write your own integration, and (if you'd
+like) to get it merged into the main Zulip repository.
+
+Integrations are one of the most important parts of a group chat tool like
+Zulip, and we are committed to making integrating with Zulip as easy as
+possible.
+
+## Set up an existing integration
+
+Most existing integrations send content from a third-party product into
+Zulip.
+
+* Search Zulip's [list of native integrations](/integrations) for the
+  third-party product. Each integration has a page describing how to set it
+  up.
+
+* Check if [Zapier](https://zapier.com/apps) has an integration with the
+  product. If it does, follow [these instructions](/integrations/doc/zapier)
+  to set it up.
+
+* Check if [IFTTT](https://ifttt.com/search) has an integration with the
+  product. If it does, follow [these instructions](/integrations/doc/ifttt)
+  to set it up.
+
+* Check if [Hubot](https://github.com/hubot-scripts) has an integration with
+  the product. If it does, follow
+  [these instructions](/integrations/doc/hubot) to set it up.
+
+* If the product can send email notifications, you can
+  [send those emails to a stream](/help/message-a-stream-by-email).
+
+## Write your own integration
+
+We've put a lot of effort into making this as easy as possible, but all of
+the options below do require some comfort writing code. If you need an
+integration and don't have an engineer on staff, reach out to
+`support@zulipchat.com` and we'll see what we can do.
+
+### Sending content into Zulip
+
+* If the third-party service supports outgoing webhooks, you likely want to
+  build an [incoming webhook integration](/api/incoming-webhooks-overview).
+
+* Otherwise, you can
+  [send messages using Zulip's API](/api/send-message).
+
+### Sending and receiving content
+
+* To react to activity inside Zulip, look at Zulip's
+  [Python framework for interactive bots](/api/running-bots) or
+  [Zulip's real-time events API](/api/get-events-from-queue).
+
+* If what you want isn't covered by the above, check out the full
+  [REST API](/api/rest). The web, mobile, desktop, and terminal apps are
+  built on top of this API, so it can do anything a human user can do. Most
+  but not all of the endpoints are documented on this site; if you need
+  something that isn't there check out Zulip's
+  [REST endpoints](https://github.com/zulip/zulip/tree/master/zproject/urls.py)
+  or email `support@zulipchat.com` and we'll help you out.

templates/zerver/api/sidebar_index.md

@@ -1,12 +1,9 @@
 ## Integrations
 
-* [Overview](/api/integration-guide)
+* [Overview](/api/integrations-overview)
-* [Existing integrations](/integrations)
+* [Incoming webhook integrations](/api/incoming-webhooks-overview)
-
+* [Hello world walkthrough](/api/webhook-walkthrough)
-## Incoming webhooks
+* [Non-webhook integrations](/api/integration-guide)
-
-* [Overview](/api/incoming-webhooks-overview)
-* [Walkthrough](/api/webhook-walkthrough)
 
 ## Interactive bots (beta)