I am using REST client in my firefox to test a connected app. My app is the sample test class from one of the tutorials services/apexrest/Cases/ that creaes a new case. In my connected app  have given callback URL as my salesforce instance appended with /services/apexrest/Cases/ . I am using OAuth 2.0.

In my REST client I have specified the client_id, secret correctly. I have provided auth URL as https://login.salesforce.com/services/oauth2/authorize and token URL as https://login.salesforce.com/services/oauth2/token. I have specified Grant type as Authorization code.
In my REST client, pressing Request Token does not provide me a token. Instead it asks me to login to my salesforce. I repeated all of these in POSTMAN as well - same result.

My objective is to be able to provide an endpoint (in salesforce) with relevant parameters to a thirdparty that can simply call my api along with these parameters and additional data that I add as a new Account in my salesforce.

What am I missing / doing wrong? I dont expect the thirdparty to be shown a login screen of salesforce  - I thought my client_id and secret key authorizes that. I was expecting to get an access token in my REST client that I would have used to post the next set of data but I cant seem to get past the first stage. Is it necessary to have an application URL to be able to test this in place of Rest client ?

Thanks in advance for any suggestions.