Is there an easy method to send the app to my phone or do I have to do it using debugging from the development computer while being connected to the phone?
Also, do I really have to sign, version, prepare and publish the application before I can actually use it on the phone? Or, can I use it on just my phone without publishing it to the world?
To use it for yourself you can send it through ADB or just copy apk file to your phone.
If I develop an application in Eclipse how do I send the app to my phone?
Click the Run toolbar button, or choose Project|Run from the main menu, IIRC.
Is there an easy method to send the app to my phone or do I have to do it using debugging from the development computer while being connected to the phone?
"using debugging from the development computer while being connected to the phone" is "an easy method to send the app to my phone".
Also, do I really have to sign, version, prepare and publish the application before I can actually use it on the phone?
All the necessary steps are handled for you by clicking Run.
Or, can I use it on just my phone without publishing it to the world?
Whether you use Eclipse or command-line tools, you can push your APK into your phone without publishing it to the Android Market.