Hi, I have Sonar X3. It is connected to a Yamaha Portable G-1. I am using a Windows 8.1 computer.
My input is set to the Yamaha (connected by USB), and my output is set to a Roland Quad-Capture.
When I play midi, my sound bars go up and down, and I can hear what I'm playing through my headphones. The midi piano roll shows up on the track where it is recording. When I try to play it back, the sound bars move, but I cannot hear anything. I have went through Preferences, and I don't know how to fix this problem.

What are your channel and patch settings in the MIDI track?
 

My output channel is set to MIDI Omni. My patch setting is on Acoustic Grand Piano.

I'm a little unsure on what you are trying to play. If you want to hear the sound from the G1 you need to make sure that the MIDI track output is set to that and then the playback is monitored from the G1 not your audio interface.
 
If you are trying to play a softsynth then make sure that "Input echo" is enabled, and the softsynth has some form of audio output, be that a separate instrument track with it's input set to the softsynth or a "Simple Instrument track" in which case the MIDI and audio are "all in one".
 
It sounds like a routing issue. Have a look at this video on my Youtube channel which covers basic MIDI routing with both external MIDI devices and soft synths.

I checked out the video. I'm pretty sure I did everything it said. My preferences are the same. My input is set to Yamaha Portable G1; my output is set to the Quad-Capture. That being said, I did try adding a patch but it still didn't play.
 
My optimal goal is to hear the actual sound from the keyboard come back through the Quad-Capture after being recorded. Is there a way to send the MIDI input from the keyboard, and have the keyboard sound be what I hear on the MIDI playback?

Oh no. So I was playing audio, which has worked on playback in the past - I thought it was just MIDI that was messing up - and there is no playback. In fact, when I tried to export, to see if I could hear it in iTunes, it said that there was no audio data. So the problem is playback, not specifically MIDI. I did not know that.