Which VOIP System’s Connect Directly to the Eithernet on My Router? What Is the Best One?

As a full home phone VoIP alternative, you can do it with services like Packet8 or Vonage. They provide everything you need but at a higher price than some others.

Ooma or Joip may be a good options to consider too.
Skype can be used directly connected to Internet when using the Linksys CIT400 dual-phone adapter.

If you have some technical and networking savvy, the cheapest option is to sign up with BYOD (bring your own device) VoIP services like CallCentric or InPhonex. That is what I do. I purchase my own Linksys SPA2102-NA VoIP adapters which plug directly into the Internet Modem. As the SPA2102-NA has its own built-in Router with QOS, no external router is required. It takes a litte technical ability to configure the SPA2102 to VoIP carriers like CallCentric, but they provide device configuration information on their web sites for the do-it-yourself people. You can purchase phone numbers in any USA/Canada area code as well as in numerous countries around the world. They are pay-as-you-go service, which means no contracts, no commitments, and no non-sense service.