Best to take it to someone who knows what they're doing, as there could be several different causes.
The neck should be almost dead straight, with the barest hint of a curve. You can test this by fretting the same string at both the first and last frets. It should be barely clear of the neck in the middle. If you fret in both places and there's a lot of space between string and neck, then the neck relief needs adjusting, which is easy if the guitar has a truss rod and more or less impossible otherwise.
It's unlikely to be the nut -- if the nut was too high, the problem would be most apparent at the first fret.
If the neck relief is OK then you can try reducing the action at the bridge, but there's a limit to how far you can take this. Don't do it by filing deeper grooves -- remove the saddle and take the extra from the back of it.
In all honesty if it's an old guitar then the chances are a neck reset may be the only option, but it's expensive.