Hi Tonyia,
First of all, congratulations on adopting a great exercise program and jumping in with gusto! And also for your signature -- eating better, drinking water, and feeling great!
Like gardenerjoy said, you will get as many different answers as people you ask... so here's my $.02.
Yes, you need a day of rest. Maybe elite athletes train 7 days a week, but we are mere mortals. I would suggest resting at least one day a week, and maybe consider a "light" day, especially if you are getting fatigued.
As far as results, what kind of results are you looking for? If you are hoping to improve your cardiovascular fitness, strength and toning, you are probably already seeing results (i.e., you can go faster on the elliptical or use more resistance, you can do more crunches, you are not sore after doing your weight work).
If you are looking for lower numbers on the scale or inches off your measurements, you will have to take eating into account. Our bodies are very good at demanding more fuel when we exercise more.
This article describes the dilemma very well (you may have to register at the NY Times website to read the article but it is free -- and they frequently have interesting health/wellness/exercise/weight loss articles). One of the 3FC members has a great signature line: "You can't out-exercise bad eating habits." That includes eating too much of even the healthiest foods.
I had a workout today like you described: I felt horribly tired and hoped my workout would perk me up, but instead it just ran me down more. I'm taking tomorrow off and will be back in action again on Wednesday!