Cable and Internet connection help

I hope you guys can help. I want to add cable to a new room I built in my basement. Also, I need to move my desktop computer downstairs. I have Verizon Fios internet, cable and phone service. I called to ask about a tech coming out to my house to do the work. THEY WANT $320!
I have added cable to a room (much better than they would do) and added a wall plate for the coax cable. Could I just use another splitter? Currently, it goes from one coming in the house to three – a three way splitter. Should I just buy a four way to add a connection?
As far as the router for internet is concerned: It is currently hard wired to my desktop. I have an iphone and ipad that I use, but for the purpose of moving my computer – can I install something on my computer to receive wireless connection. That way I can leave the Verizon router where it is.
Please help!