The idea of TDD is to code only what you really need and is a form of Test First. It's really a way of nailing down acceptance criteria for a user story more than it is a way to unit test everything that goes out.
It doesn't really sound like this is what you're interested in to me, at least not at this point. Rather you may be looking at some more ...
Advantages of TDD:
It should reduce the number of bugs and hence save on bug-fixing time
It gives developers more confidence to refactor
It can help get a better design (thinking about testability helps with design)
Disadvantages of TDD:
Extra effort in writing the tests
When refactoring, the tests may also need to be refactored, complicating the process
TDD is more about how you write production code. If you need to write a large pice of code TDD allows you to split the work into small pieces. And tests guide you all the way. There are many cases when this approach leads to faster development, but there are also some cases when it doesn't (if you have too much doubts about the design of your code you may ...
To echo what the others have hinted at:
Testing is not a necessary evil.
Testing is a crucial component of delivering quality (or at least working) software.
Keep in mind that you only get one chance to make a first impression and delivering buggy software destroys your reputation.
So, to answer your question: You need to budget in more time for testing - ...
The real problem here is that your organization is framing the cost of testing as unwanted overhead, rather than as an expected and necessary cost of doing business for product development, maintenance, and support. Focus on fixing that.
Analysis and Recommendations for Reframing
Would you have any tips on how to improve this process so that testing ...
My devs are not used to unit testing framework and are still learning
This will improve over time as they gain more experience.
There are only 2 of them and each with their specialty (Front-end and back-end) so they have to do both the coding and the unit testing for their side of the stories.
That is absolutely normal. Who else would write tests? Even if ...