Two things of note that may explain the present issue/s.
The computer is relatively old so there may not be any Windows 10 drivers available.
The computer did not ship with onboard wireless capabilities which suggests that a USB type wireless device or a PCI-E wireless card has been used.
Upgrading from Windows 7 to Windows 10 never goes without the user having problems at the best of times but the problems are always worse with older hardware down to the fact that there are no Windows 10 drivers available.
i suggest that you download a Windows 10 ISO and whatever drivers we can find and then do a clean install of Windows 10 and the drivers, before we provide the info for the Windows 10 ISO you need to do a couple of things first.
Let us know the wireless device details so we can look for the drivers.
Check what drivers are available to you at the emachines download centre here
You are welcome btw