Do you have a soundcard besides the Delta in that machine? If so, you need to go to the Sounds icon in the Control Panel, then the Audio tab. Make sure that you have "M Audio Delta 66 Multichannel" is selected in the Sound Playback and Sound Recording sections.
If that is right, the likely problem is in the Delta Control Panel. Have you read the Delta manual (the blue one)? Check out page 13. Getting your head around the Patchbay Router is not easy; I still get confused.
Plug a pair of headphones into one of the Phones jacks on the front of the Omni and turn its knob up to 4. Start playing a CD in your CDROM and open the Delta Panel. You should see activity under WavOut 1 and 2. Uncheck the Mute box for both, if necessary, and raise their faders up to max (0 db).
Now go to the Patchbay panel and look at the left column, H/W Out 1/2. This control what is sent to the card's outs (monitors and phones). Since the activity is on WavOut 1/2, check that button. Or you can check the Monitor button, if you want to use the Monitor faders to control levels. Go back to the Monitor Mixer pane, and make sure the Monitor channel is un-muted and the faders are up.
As for which jacks to use for your speakers, check the Omni manual (the red one). On page 6 it shows you the back panel. The Monitor Outs accept either TS or TRS, so the 1/8 to 1/4 adapters you already have should do the trick. Note that if you have a super cheapo set of speakers with only one 1/8 plug, you are not likely to get stereo output by putting that plug into the L jack. You should get sound, though.
Good luck,
wren