Hi We need API (REST/SOAP) web services to PULL /PUSH process and t

Hello,
The default Call Web Services node in an out-of-the-box (OOTB) Appian installation uses SOAP web services. You can configure and use that node to make web service calls from Appian.
Please take a look at:
forum.appian.com/.../Call_Web_Service_Smart_Service
for more information.

We are looking at services to access to access task and processes from Our system
For Example :

1.Submitting the task from webservice by giving input data.
2.Getting List of task Ids.
3.Getting process related task Ids Etc.

We are looking at a service layer where we can work similar to tempo/apps view.

The Call Web Service node only allows Appian to make calls to other SOAP based web services. There are additional web service components available under the Shared Components section on Appian Forum to make calls using other web services than SOAP.
forum.appian.com/.../all

You may wish to investigate the creation of some custom smart services to address your needs. Please see the documentation on custom smart services able to created for use within Appian at:
forum.appian.com/.../Appian_Plug-ins
forum.appian.com/.../Custom_Smart_Service_Plug-ins
forum.appian.com/.../Custom_Servlet_Plug-ins

So We have to write custom plugins to have Web-services as per our requirement and Appian wont providing any Web-services to work with task and process apart from exposing process model as Web service . Am i correct?

That is correct. Appian contains the functionality to call SOAP web services via the OOTB Call Web Service node, the ability to call other web services using Shared Components available on Appian Forum, and the ability to trigger process models by Publishing a Process Model as a Web Service.
forum.appian.com/.../Publishing_Process_Models_as_Web_Services