Is my camera going to work with these lenses?

I have a Nikon D5000. I got it about a year and a half ago as my first DSLR. I really liked it, but now I'm wanting to upgrade my lenses and the more i read the more it seems that the professional quality lenses like f2.8 wide to telephoto or just telephoto zooms don't focus correctly with it.
I found with my 28-75 it keeps only focusing part of the image. Then i read that it is because of my camera type… Is there truth to this? I know it is an amateur camera and there are a couple things that bug me about it, but i thought i could at least keep using it until i got some paying customers!
Any helpful info is appreciated.

Added (1). Sorry, it is a tamron 28-75 f2.8
i know my camera does not have a af motor in it

No info on which 28-75= no helpful info

Nikon works best with Nikkor lenses.