Nick here from Shopify. Really good questions!
Taxes can be difficult to figure out even on the best of days. The first thing I always suggest to someone who is not sure (and even to people who are sure) about their taxes for their store is to reach out to a professional accountant, tax expert or your local tax office. The reason for this is that taxes are different for every country and even specific regions within countries around the world. What the taxes might be in Ireland are completely different to the taxes in the USA for example.
I'm not sure what country you are in, but to report to the revenue commission/ tax authorities/government you would normally need to register the name of your business name with your local tax authority. This is something which you could do online I'm sure, but again, it is something I would recommend asking a professional about. I actually went through this process recently and reached out to a professional accountant. It has made the process much easier for me and taken a huge weight off my shoulders. To not include tax in your pricing could have legal issues which would depend and differ based on your location. It's not something I would suggest doing personally. For example, it could result in your business needing an audit, where you would have to pay for the auditor which can be quite costly.
Shopify has some very helpful guides on how to set up taxes for your store. I'm not sure where you are based but there are guides for taxes if you are based in the USA and Canada as well as taxes for if you are outside of the USA and Canada here.
I hope this helps and answers your question as well as giving you the best next steps to take here. Let me know should you have any other questions on this.
All the best, Nick