Many roof leaks are quick to fix, you just need to know what you are looking for and have the right tools to fix the problem.
Perhaps the most common problem with a roof is a broken or slipped slate. Over time slates can degrade and any that have minor imperfections can crack. The nails which hold these slates in place can also deteriorate and actually break where they hold the tile to the batten below, causing the offending slate to slip.
First use binoculars or similar to inspect the roof from the outside. If the leak is isolated you may be able to spot the offending slate from the outside before scaling the roof to replace a slate. If you cannot see any obvious damaged slate go back inside and, in your loft go to the point of the leak and follow the moisture back up to the source. You may need to remove any boards covering the rafters before looking for discolouration caused by water or any mold that may have already started.
Once you have located the highest point of the water ingress, you can remove the breathable membrane to find the exact source of the leak.
If you can access your roof safely and get to the problem, replacing a slate is very easy. If the slate is being held by nails, you'll need to use a slate ripper to cut through the nails under the slates above. You should then be able to pull the damaged slate downwards and remove it from the roof. You can then use a metal slate strip to fix a new tile in place, negating the need to nail the new tile in place.
When you are working on a roof, it's a good idea to make sure you take precautions to get your tools and roofing supplies on the roof in a safe manner and lower tools and roofing supplies to the ground by the same method and a bucket and rope arrangement can work well here.