You can set it up yourselves, very easy, check cro.ie to get started, no Accountant or Solr required for this stage, I would recommend a shareholders agreement after formation which will guide you as to Directors roles, responsibilities, I would use a Solr for this.
I'd suggest you use a company formation agent to do this. They charge anywhere between €200 and €700 (for the same product) and use Google to find these. As it mentioned you can do it yourself but it is rather tedious and using a formation agent will be much easier and quicker. I probably get ten companies formed a year and wouldn't think for a moment of doing it myself.
A "trading name" or Business name is used if the company name is not what you want to call your business. The Company Registration Office site has full details here https://www.cro.ie/Publications/Overview