What should a bass sound like?

It should sound like an oiltanker dragged across concrete :)

Heh, a Finnish band who has recently hit it big over here used a similar description when recording their debut album. They're from the countryside and told the engineer they wanted the bass guitar to sound like an old tractor pulling a sheet of metal on an old sideroad. Granted, the bass sound on that record is pretty awesome :)
 
Here is what I do with the bass when tracking :

Using new strings for a recording is very important and you will get a lot more depth and thickness to the tone.
I usually use a splitter, the important thing is to get one clean signal, and one distorded (amps or DI, depends on what you can use, amps are better really).

When mixing I usually do A LOT of filtering to see what happens with each signal recorded.
I usually highpass ther distorsion @ around 500-700 Hz, and lowpass @ 2-5 khz. Keep only the clear midrange agression. Maybe you'll have to boost 2 or 3 db of 800 Hz to make the growl appear and cutting the mix. Sometimes you'll have too much of 800 Hz (nasal sound) and you'll have to cut there. I also like a slight boost around 2-2,5 khz too.
Then you have the clean track, it can be a nightmare. I generally highpass it around 60 Hz and lowpass it somewhere between 150 and 300 Hz, depending on the clarity of the low mids area.
From there I generally have to cut some 250 Hz to make things clearer, and then I check the area between 100 and 200 Hz which is very important and has to compliment the low mids of the guitars. Some slight EQing here and here can help to get the good amount of low mids, they do not have to fight with the guitars fatness.
Then comes the "low" part of the bass. Grab an EQ, boost 10 db somewhere between 70 and 90 Hz and see where is the key frequency for the bass to sit in the whole mix. I find that it's generally 75 Hz or something. Reduce your boosting to taste (you'll need a sub or good headphones to check that). When you think it's good, that boost in the lows of your bass clean track should sit just behind the lows of the guitars (yeah the 90 Hz boost you applied to make them bigger).
Then send those two tracks (disto+clean) to a bus that you can compress (usually I do something like 3:1, fast to medium attack, fast to medium release, threshold -7 db). Hope this can help you a bit.