Here is the way I usually go about rendering fur in Photoshop. You'll find that its actually pretty simple, but can often require a lot of patience if you want to make it look realistic.

Create a custom brush consisting of a few multi-sized dots. The dot sizes will vary depending on the resolution you're working at. Make sure you don't use very many dots either, or the pattern will be too repetitive and noticable. Once you've created the brush, set the spacing to "1" by double clicking it in your brushes menu. I have several brushes that I use, but here are two examples of how they look... feel free to use them for yourself.

Step 2 - Choose a fairly neutral base color to start with. Set your foreground color to a darker variation on your background and begin to paint in short sweeping strokes. If you're doing this with a mouse, you might want to make the brush "fade" out since you don't have pressure sensitivity. Don't worry about it being a little messy. :)

Step 3 - With a lighter color, paint more strokes over the original image. You'll find that fur consists of many different colors. You may look at your dog and think "brown", but get up really close and you'll see there is a lot of variation there. Also keep in mind that the fur more than likely won't be rigid and straight, be soft and fluid with your painting.

Step 4 - In this step I've gone back and added more dark fur, and also painted more light fur across the top. I'm based this off of my cat, who has a patch of white froof right under her chin - that is what I'm working towards. :) I also used a deep golden color in some of the darker areas to bring out the highlights.

Step 5 - Ok... now that we have a pretty good base down, we can go in and sharpen up some individual hairs. I use this step to cover up some of the messier areas of the background and to bring out the really bright strands of hair. Use a small round brush and start drawing individual wispies. If you need to, go back over with the smudge tool to smooth them out. Often hair lays down in clumps, do you'll have areas of shadow beneath sections of fur.