I'm a guitarist who has recently taken up bass too. I already know that with guitar, the lower the action the better (with as little string buzz as possible).

I'm not sure about bass string height. I was told that a certain amount of buzz on a bass fretboard is acceptable. So how high should I aim to adjust my strings?
Well, as low as you want it, whilst also minimising the amount of buzz you get.

So whatever you're comfortable with really.
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I want as low as possible for fast easy playing. I just don't want to hear complaints that there's a ****load of buzzing out of my bass and it's too late to adjust it.
Fret buzz is a lot more where the fingers are than the height of the string. Make sure your fingers are as close to the fret as possible, but not on top of it.

String height does make a difference however, and if you have a light touch, you can keep the strings lower, but if you really want to dig into your bass, it is best to have them higher to avoid fret buzz.

If you want to avoid fretbuzz altogether, get a fretless, however, unless you have a lined fretless, it is very easy to go flat or sharp.
Personally I prefer kinda high action, but high enough so that you get a bit of rattle when you dig in.
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EDIT: Sammcl pretty much got it dead on.
i believe 2.5mm on the E string from the 8th fret
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Is there such a thing as a heavy riff with out chugging on the e string?
When you first start, as long as its not hurting your fingers you can have em high enough where they dont buzz, all though Frobot is very right in his comment. As you improve though you will find that slap, tapping, and all other teqniques you will want to lower it some so you can get the various string aginst the fretboard sounds. But alot of it is in the srings too.
+1 for Collin.

I wanted to mention the technique stuff, like tapping and slapping etc. It is much easier to get a good percusive sound when the strings are lower.

Going further into what Collin said, strings play a roll as well. Roundwounds will be brighter and have more treble/fuzz/rattle to them when digging in, while Flatwounds are more mellowed out.