Nice shots, they make good use of the low aperture.
as said, your white balance is off, I found leaving it on daylight works well for most situations, but auto is incredibly good in new cameras so make use of the auto white balance feature, but as shown, it isn't always correct.
a good thing to do when your starting out, use manual mode (M) and do not use AUTO, I feel once you use auto, you learn bad habits, if you limit yourself to x amount of shots using manual (limit to about 36 shots per session <thats how many exposures are on a typical film roll> so you have less chances of getting a fluke.)
and try get a good exposure (and for bonus points, composition!)
This should help you greatly, my camera is almost always in manual, but not every shot will be a correct exposure, but that is okay, you are only starting out.
The best way to tell if your exposure is correct is by using the histogram!
I am not sure how to display the histogram with the nikon D5000 but this is a link to how to use the histogram.
How to Use Histograms
also in shot #2 its a bit blurry, try and learn to hold your camera still,
the best way to do this I find is to try take a picture at the longest exposure you can (start at 1/15 then drop to 1/6 then to 0.5" then to about 2" and just try get it sharp as possible at 2" it might be a bit hard, but there is technique to it (if you use your camera strap it helps, there is some videos on youtube you can look at by searching "how to hold a dslr steady"
[ame]http://www.youtube.com/watch?v=YJGYShZNDYY[/ame] notice how his strap is wrapped around his arm.
another tip I feel is useful...
After you get manual mode down, or when out shooting something important (assuming you are still learning manual) use the semi auto modes (A and S)
I prefer using the A (aperture) semi-auto mode when my shutter speed doesn't matter, however if I am shooting something that I need a certain shutter speed I use Shutter priority (S)
as I say though, take your time and learn no need to rush into it.
(and ofcourse if your not looking to take anything fancy like landscape or w/e) the shots your taking just now are great.
I will leave it at that and good luck

enjoy your new purchase.