What I am trying to achieve is secured wired network and an "open" wireless network with a single internet connection
I have the secured part (a dlink DI-624 with the wireless turned off - it is connected via ethernet to 2 computers) but what is the quick and easy way to add the wireless part of the equation (for the sole purpose of having roaming internet access around the house) without comprimising the secured wired element? Can I just plug a wireless router to the existing router?
I'm assuming by "secure" you mean limited access, because access to your network is limited only to wired connections.
Yes, you can connect a wireless router to the existing router. Unfortunately, unless you place a password on the hotspot you'll create with the wireless router, people will be able to access your current "secure" network, as everything resolves to one router.
Your best bet is to secure your wireless router with WPA, then just give it to friends, loved ones, and other truested users who happen to visit your area.