Did anyone make the point yet that walking connotes quarter notes? That is, the walking line matches the tempo pulse. If you count off the song, "1...2....1-2-3-4...", then the walking line is going to be at the tempo of the "1-2-3-4" counting (sorry if I'm condescending here--just trying to communicate the idea of a quarter note if that's not clear here--no elitism intended! I'm here to

).
So, walking line=quarter notes (with some exceptions granted on occasional fills, grace notes, etc.). It's not just the notes themselves, but the rhythm, which is more or less constant.
And as for note choice, check out some FAQs here on TB if you haven't already:
http://www.talkbass.com/wiki/index.php/Walk_With_Me_-_Walking_Bass_Lines
Basically, you have chord tones, scale tones, and chromatic tones. In really general terms, those are ranked in order of tension. Chord tones over a that chord create the least tension. Scale tones create more "movement" or desire for change. Chromatic tones--usually a half-step away from the next chord's root--create lots of desire for resolution, and thus, there's a lot of movement and tension created by chromatics.
A way to start to think about these things is that on strong beats, you use relatively low-tension notes. On weak beats, you use tension to help move things towards the next strong beat. This is really general, and there are exceptions, but "walk before you run." (pun intended!)
An example of
four beats of C going to F:
C - E - G - Gb | F
That's the root, C, on beat 1, the strongest beat. Next, there's an E, the third, on beat 2. There's little tension there, but we're not changing chords on beat 3. Then, on beat 3, the G, which is almost as boring (low-tension) as the root, but that's fine on a strong beat like 3. Then, the Gb, which has very little to do with the C chord, and thus, there's dissonance ---> tension. This is good on the weak fourth beat moving to another chord on beat 1, measure 2. Plus, the G - Gb - F movement is in half steps, which is very strong movement.
[DEL]It's almost a cardinal rule that you always play the root of the chord on the beat in which the chord starts. There are rare exceptions. In our example above, you'd almost always play 'C' on beat 1, measure 1, and 'F' on beat 1, measure 2.[/DEL]
This is just one example of how to approach some of this stuff. The Wiki is great material to look over, understand, and practice.
Happy practicing.