The basics:

Each finger (generally) gets in own fret. If you're in first position (index finger at the first fret), your middle finger plays the 2nd fret, ring plays the 3rd, and pinky plays the 4th. 2nd postion means index is on 2nd fret, middle on 3rd, ring on 4th, pinky on 5th. and so forth. Ideally, you want the finger pretty much right on the fret, or just slightly behind it (like fraction of a fraction of an inch behind). This way if you ever play fretless your intonation will be pretty good.

This can be tricky on folks with small hands (especially in lower positions), and I advocate pivoting off the thumb a bit to make reaching the right fret a bit easier. Above all, if you experience pain, STOP. You are probably holding too much tension in your wrist and fingers, try to relax. The strange catch 22 about this is, you need to hold down the string with a certain amount of force to get good tone... I try to think of it as being the weight of my arm holding down the string, not the strength of my fingers (although that is a part of it).

A key aspect is economy of motion. Don't move your fingers that aren't fretting a note off the strings, or at least not far off.... I went through a phase where people couldn't see my hand move at all, but heard a great variety of notes. The closer you are to the note already, the easier it is to get it to come out at the right time.

It's good to practice independence of the ring and pinky fingers, try setting a metronome and playing 8th notes to it (two notes for every click) downbeat on ring finger note (anywhere on the neck) and upbeat on pinky note (one fret higher). Keep your index, middle and ring fingers down on the string, and just move the pinky. Also try the same exercise with the downbeat on the pinky note. Also try putting the pinky on another string (and keep it down) and alternate between a ring finger note and a middle finger note, this time only moving the ring finger.

A few more ideas... examine some more "guitar" oriented techniques like bending, vibrato, trills (rapid hammerons and pulloffs between two notes with only an initial attack from the right hand), what I call a "shake", where you fret a note, and slide the fretting finger around to adjacent frets quickly while letting the string ring, I mentioned in my above convo with mistermaxxx how Verdine White uses this technique to great effect.

One that I used to death when I first discovered it, and am now trying to incorporate more artistically is the grace note from below. Grace notes are nothing new, but the way I approach them on bass is a bit different (not that I invented it, someone else surely did, but I never read about anybody else doing it this way). If I'm going to play a D on the A string, I might attack the note as a C and hammeron quickly to the D so that the note is heard as a D with a little "dip" leading up to it. You can do this from above as well, but it sounds more obvious, and the note you fret will be heard as a part of the line. This is good when playing a line that may have originally been done on a synth, as synth sounds tend to have some sort of complex attack that may include pitch modulation (usually it's a filter opening). Alternating between a straight attacked note and a grace note like this in the right place can be the difference between a line that's just there and a line that comes to life.