It's a steel-string, and I've just discovered that there is fret buzz starting from the 1st fret

Previously, I've put a bit of paper under the string at the nut, on my electric. But that was buzzing on the open string. How can I sort it out for this? If I take it to the shop, how expensive/long will it take?

does the bit of paper stop the buzz? if so, you may either need a new nut or to have your neck relief adjusted.