Automated testing in web applications

We are developing a very large web application using C# and JavaScript. The majority of our calls in the code are handled with JavaScript on the front-end and the only real back-end work is the SQL server which retrieves the relevant dataset and returns it as a JSON object which is then handled once again with JavaScript.

We are currently doing all testing manually with our testers who verify the page functionality against the design specifications to make sure the software is working as expected.

I would love to be able to add some automated testing to the application. I've looked into Unit Testing and it seems that it works a lot better if you have a lot of C# methods with inputs and outputs which are easy to trace and verify that they are working.

What are the standard methods in use in the industry which we can use to verify that the whole application (or at least a majority of it) is working properly?

The sort of things I want to check are:

1. We have an input page which lets you create a user. This input page should update, let's say, 5 tables in the database. I am adding 10 different pages which allow me to create users in the application, how can I verify that all of those pages are inserting correctly and working properly?

2. I have a page where I can click a button which inserts a row into the database. How can I check that it's outputting correctly in all the different places it should output?

Off the top of my head, I cannot think of all the different cases I need to check but there are a huge number of them. As far as I have understood, any visual errors can only be tested by users who are manually testing.

I am looking for some feedback on what the best methods are and also how they can be applied to our type of application.