You should go first with a mobile application or a website?

Responsive Web site.

 

Before deciding whether you have to start with a website, an application or both I would find answers to the following points first.

1- Have you done your research and identified the differentiation of your app?
2- Have you identified your demographics (their age, their geographic regions, etc)
3- Have you done any risk analysis?
4- Have you identified whether the social media use in the geographic countries of preference is high to assess if your communication/launch strategy will achieve the required market penetration?
5- Have you assessed if the competition could claim the idea taken from them?
6- Did you patent the idea before discussing it with any internal or external stakeholder so no one can copy it and implement it faster?

There is much more questions that need to be answered in order to build the business case.
Only then you can move into discussing the brand identity development, the platforms that you want to use and the methods, tools and communication channels you will need to employ for the purpose.

A detailed research will give you all the answers including whether the investment is worthwhile and whether the idea has a chance in the middle of the fierce competition.

Lastly if you have idea and create a website. Make a site responsive so that it works for all devices, and there are also ways to use site as an application.