I understand that you have some questions about the Slingbox and the first time set up.
The Slingbox must be connected to your home network. It must be connected to the home router (which will provide the IP address the Slingbox needs to reach the Internet) and also to your TV source (cable box, DVR or satellite receiver). Once you set these connections, you will run the Setup Assistant for Internet Viewing.
When at remote locations, you need to go to www.slingbox.com, log in to your Sling Account and hit "Watch". So, it is mandatory to connect the Slingbox to the home router, otherwise you will not be able to set it up for Internet Viewing. The following links will provide you with further details / information in order to get your Slingbox SOLO up and running.
Finally, you can run the whole process using a computer that meets the requirements for the Slingbox SOLO. It does not matter if you use a desktop or your laptop computer for the first time setup. Once you finish this process, you can take your laptop with you in order to get the remote connection to your Slingbox. This link will allow you to check if your network and computer meet the requirements for the Slingbox SOLO.