Fractional First Officer reported there was confusion on the signage to the FBO and the controller also spoke too fast when providing taxi instructions; which led to a taxiway incursion.

Narrative

The Captain and I briefed we would land Runway 22 and make a left turn on M or J to get to the FBO. After landing Runway 22 Tower told us to exit left onto Runway 31L and then contact Ground. I contacted Ground and told him we were on Runway 31L for the FBO. He then rattled off taxi directions which I believe he said it was make a left on Lima cross Runway 13L Charlie Papa. I paused since I was looking at Runway 31L and R. I asked for him to verify taxi instructions. since I was sure about crossing the runway and getting to Papa taxiway. He repeated it fast and the Captain said he got it so we proceeded on Lima and turned on Mike. I told the Captain to hold since I wasn't sure about being cleared to cross 31R at Papa. I called again to confirm if we were cleared to cross the closed runway 31R at P. The controller informed us we were cleared to cross 31R at Lima then Charlie and Papa to the FBO but the Captain made the short turn on Mike to Papa to cross since it looked like Papa signage was confusing. So the Ground Controller asked if I could hear him and I said yes sir are we cleared to cross Runway 31R on Papa which he then said cleared to cross Runway 31R at Papa to the FBO. We shouldn't have turned on Mike to Papa but kept straight ahead to Charlie so I'm glad I stopped us right away to confirm the rest of the route.Suggestions: There seemed to be some confusion on the signage to the FBO. I should have been more assertive with the controller to slow down the taxi instructions and repeat them back again and write them down again. When he turned on Mike taxiway I told the Captain to stop which he did and I contacted the controller right away to ask about our taxi route and clearance to cross 31R which was closed. He then cleared us across the runway and taxi Papa to the FBO.