If you are getting the Genie, you have a few options. If you get the HR34, you can connect an Ethernet cable directly to the HR34 and that will get you Internet to ALL of your receivers. If you get an HR44, you also have the option of connecting back to your router via its built-in wireless option. If you don't have Ethernet near the Genie and don't get a HR44, you can always get a Cinema Connection Kit, which is about $25 from DirecTV and sometimes even cheaper via eBay or Amazon. The CCK would be installed on a coax connected back to the dish and have an Ethernet cable connected to it going back to your router. There is a wireless-CCK, which can connect back to your router via wireless, however, it is not installed very much anymore.
If you are having a tech come out to do the upgrade, he will make sure that you are connected to the Internet. If you don't have a coax near your router now, the upgrade includes the install of an additional coax run to that location as long as a HR44 is not being installed (as they would use the HR44's wireless then).
Let us know what your current setup is and what you will be getting and we can probably be a little more specific in regards to how things will work for you.
BTW, welcome to DBSTalk!
Today's problems don't worry me, I haven't solved yesterday's yet.
SlimLine-3 Dish w/ SWM16 (HD Service / WHDVR) / Full Setup
HR34-700 / Panasonic TC-P50G25 HDTV / HDMI / Networked - DECA / Family Room
HR44-700 / Samsung HCM5525W HDTV / Component / Networked - DECA / Bedroom
HR24-100 / Samsung HCM5525W HDTV / Component / Networked - DECA / Bedroom