Absolutely 110% backing up Tiny's last comment regarding the eyes. Bugger up the eyes, and you've wrecked the portrait. If the eye's aren't right, it will look like any old horse/dog/cat etc., get the eyes right, and you can make small mistakes in the rest of the portrait but get away with it, as the focus will be pulled to the eye.
I always do the eyes first when I start a portrait. As tempting as it is to block in all of the colour and start the bulk of it first, you will waste a lot of time if you botch the eyes and have to start over. Because I work in soft pastels on a velour backing, having to start over after already having done the bulk of the subject, would be both expensive and time consuming. So I always get the eyes right, right off the bat :)
As well as getting the lines perfect on the eyes, you need to be very aware of toning. The eyes have tonal values from 1 all the way to 10 - use them and include them! ALWAYS!
Usually you will have a 10 (black) value along the bottom edge of the eye and the pupil, so start there, mark in your 1 value (white) and shade in the rest around those points. The eye has to look round, and it has to look wet. That's where your tones will help you. Practice by drawing sphere's with the light shining from different angles. If you can draw a perfect sphere with smooth tones, you can draw an eye.
As for fur detail, if you are a detail crazy perfectionist like myself, fur probably drives you up the wall and takes hours. If you have a close up of a subject and HAVE to draw the hairs in, think about drawing the shadows between the hairs, rather than the hairs themselves. Hairs are visable due to the light and shadow surrounding them - so draw in the shadows, and then add a little gradual tone to their hairs, letting the light do the work for you.