[Serve][Doc] Refactor the Ray Serve API doc (ray-project#32307)

- Add an index page to list all the APIs. (https://ray--32307.org.readthedocs.build/en/32307/serve/api/index.html)
- With this change, when you search specific python API e.g`ray.serve.run`. The search result will show core api link page. (Previously, the user can't get the correct search result, because we put all APIs on one page.)
<img width="604" alt="image" src="https://user-images.githubusercontent.com/6515354/217628692-720b9344-061d-44de-bc77-ee0c0ef27276.png">

doc/source/_toc.yml

@@ -315,7 +315,7 @@ parts:
                   - file: serve/tutorials/deployment-graph-patterns/linear_pipeline
                   - file: serve/tutorials/deployment-graph-patterns/branching_input
                   - file: serve/tutorials/deployment-graph-patterns/conditional
-          - file: serve/package-ref
+          - file: serve/api/index
 
       - file: rllib/index
         title: Ray RLlib

doc/source/ray-references/api.rst

@@ -9,7 +9,7 @@ API References
     ../data/api/api.rst
     ../train/api.rst
     ../tune/api_docs/overview.rst
-    ../serve/package-ref.rst
+    ../serve/api/index.rst
     ../rllib/package_ref/index.rst
     ../workflows/api/api.rst
     ../cluster/package-overview.rst

doc/source/serve/api/index.md

@@ -0,0 +1,12 @@
+(serve-api)=
+# Ray Serve API
+
+```{toctree}
+:maxdepth: '-1'
+
+python_api
+rest_api
+serve_cli
+```
+
+

doc/source/serve/api/python_api.md

@@ -0,0 +1,49 @@
+# Ray Serve Python API
+
+(core-apis)=
+
+```{eval-rst}
+.. currentmodule:: ray
+```
+
+## Core APIs
+
+```{eval-rst}
+.. autosummary::
+   :toctree: doc/
+
+   serve.run
+   serve.start
+   serve.shutdown
+   serve.delete
+```
+
+(servehandle-api)=
+## ServeHandle API
+
+```{eval-rst}
+.. autosummary::
+   :toctree: doc/
+
+   serve.handle.RayServeHandle
+   serve.handle.RayServeHandle.remote
+   serve.handle.RayServeHandle.options
+```
+
+## Batching Requests
+
+```{eval-rst}
+.. autosummary::
+   :toctree: doc/
+
+   serve.batch
+```
+
+## Deployment Graph APIs
+
+```{eval-rst}
+.. autosummary::
+   :toctree: doc/
+
+   serve.api.build
+```