|The term "home network" implies you have more than one PC or networkable system such as Game console, Network printer, NAS storage device.|
The way forward for you depends on what equipment you already have and the type of internet connection you currently use.
EG Do you have cable/DSL internet and what kind of modem? Is it a combination modem/router or do you have a separate router?
If so is it a wireless router? If you have a wired only router you can either replace it with a wireless unit ( My recommendation would be one with both wireless and inbuilt 4 port wired switch), or a wireless accesspoint that plugs into a LAN port of your existing router.
Now to your question on whether you need a wired server? No but it is generally necessary to use a wired connection to the router in order to initially set up your wireless network credentials and security protocol. You can switch to all wireless once this is done but if you need to access the router's admin interface in future this should always be done from a wired connection
Goin' Fishin' (Some day)