Hi Brandon...
Darrel's answer is pretty close here. You can certainly connect into the water pipes from the kitchen sink, but more than likely you can just connect into the water pipes from the basement. You will want to install a single lever shutoff if possible (about $30.00...see picture).
You can also connect into a 2" copper pipe in the basement (if possible), transition over to PVC (or ABS plastic) pipe and fittings and install a new ptrap, standpipe and
VENT upstairs.
In terms of plumbing code you will need to vent the washing machine or you can be sure the waste line will not drain properly or could siphon and let sewer gasses into the home. Here, you can run an 1.5" vent from the new washer and connect it to another 1.5" vent or penetrate the roof by increasing the 1.5" vent to 2"...
OR, although probably not legal in your area, you may be able to install a mechanical vent...called an
AAV (air admittance valve)...see pictures.
In terms of connecting into the pipes in the basement you will be best to see if you can connect into a 2" drain pipe. They make transition clamps for this...called
shielded clamps (see picture). Here, you can transition between copper pipe to plastic pipe using the clamps. Using the clamps you can install a
wye fitting into a horizontal waste pipe or a
sanitary tee fitting into a vertical waste pipe....let us know if you need more info. here. If you cut into any pipes be sure to support the pipes before cutting...;)
Finally, you may want to install a washing machine pan in case of overflow, but if you use the single lever shutoff valve after each load of laundry you shouldn't really need this.
SS hoses are also a good idea.
MARK