What type of ethernet cat5e, cat6, should I use to hardwire my home to use w/ Nighthawk AC1900 R7000

Cat5e will work, but I'd suggest using cat6.  You'll also want cat6 jacks.  

If this cable is running between floors, then I recommend getting plenum grade cable  (Riser might be enough - depending on the details of the cable run and your zoning.  But plenum can always be substituted for riser).  The plenum/riser cable meets fire safety codes that regular cable doesn't.  

Of course if some of it is running outdoors, you'll want an outdoor grade for that segment (and you can find outdoor plenum - which you can run anywhere).  Riser/Plenum/Outdoor cable does tend to be thicker and stiffer then normal patch cable.  But it important to meet the fire codes.

You'll want wall plates in the various rooms.  You can terminate the cables in switch boxes+wall plates at the router, or get a patch panel. Either way, you then use patch ethernet cord to connect the wall plates/patch panel to the router and switches.  There are several kits that allow you to build your own wallplate with various connectors (a mix of coax, etherent, phone line, etc).  If you are putting in a lot of these, you might also want to invest in a LAN cable tester (or see if you can borrow one).  There are some inexpensive ones out there.  A punchdown tool will also be needed (some of the wall plate kits include these).

It can be challenging to get the cables through the walls, especially if you have multiple stories.  So you might get a quote from an electrician.  Note there are some countries where it is not legal to install in-wall cable yourself.