First hospital where you can use research is at Mitton University in the second county you unlock.
Here you will be introduced to how it works.
When you have unlocked the research room it will be available at all your started hospitals and what you research will be available at all hospitals too.
As with all other skills you can train and improve the doctors skill in research which of course speeds up the research and as all other rooms you can upgrade it to level 5 which will make the researching go faster.
You have to set the room up for more than one employee if you wish to have more researchers just like with wards, you click
on the room then
then slide down to the extra staff option and click the + to add up to 4 doctors.