I have a slow-filling toilet, and it looks like a common fix is to clean out the fill valve. For some types of fill valve you just push down and turn to take it apart; this one, apparently an R&T A1250 or similar, I don't know how to take apart. I'm on the verge of giving in and calling the plumber, but if one of you fine folk happen to know what to do I may try doing it myself. Here's a picture of it.
I'd think if you called a plumber, they'd just replace the valve. I've fixed my fair share of toilets and I've never 'cleaned' and then re-installed a fill valve; there's no way to back that up. You'd get to call me back for free to do what I should've done in the first place; replace it.
That is of course, assuming the shut-off valve and the line isn't clogged. You could try closing it to squish any crud that may be blocking flow. If you do, you should disconnect it from the toilet and bleed it into a bucket while you open and close it a few times (that's what you should be doing if you have to replace the fill valve anyway).