Okay, here's the deal...the Nikon D5000 has no autofocus motor in the body. So...if you're going to shoot auto (i.e.: let the camera do the focusing, shutter speed, and aperture for you), then you need to have a lens on the camera that has an autofocus motor in it.
It's not true that most lens aren't compatible with your D5000. If you're comfortable shooting manually (where you do the focusing) than any lens with a Nikon mount (there are hundreds out there) will work for you just fine.
If you want auto capability, then just buy a lens with an AF-S designation. Or bring your body to the camera store (if you're buying it in person), try the lens on your body and see if it focuses (if you don't trust the clerk to tell you if it has an autofocus motor in the body).
Now, as to what type of lens to shoot with....I personally love prime lens (i.e.: not a zoom lens). They're sharper then a zoom lens of comparable price (fewer working parts). They will usually have a bigger aperture (that f-number thingy) that allows you to get very pleasing portraits or even some cool macro shots. However, if you don't want to swap lens and are probably going to shoot some portraits/close-ups of your daughter at play in the orchards, some landscapes, some of the baskets of fruit, than a zoom that goes out to at least 120mm would be good.
Personally, I'd suggest you look at the Nikon 35mm f1.8 lens. It will shoot some lovely landscapes (wide angle). You can shoot nice low-light photos (b/c of the wide open aperture). It's cheap. It's light. It's sharp. And it's AF-S so it will work just fine on your D5000. You can buy it new off of Amazon for less than $200.
The other thing (since you'll be shooting outside) is: get thee a circular polarizer to help your colors show their true beauty during foliage season. And do not set your white balance to auto or daylight (even though you're shooting outside in the daylight). I'll spare you the detailed explanation of camera white balance except to say that an auto setting reduces the extremes (basically it pulls everything towards the middle) so it takes those extreme reds and yellows in the leaves and it mutes them...turns them in to paler versions of what you see.