As far as cleaning up the fingerboard goes, I always used either Lemon Oil or Dr Duck's Axe Wax.
Once you have all the strings off, just apply some Lemon oil to a rag and just gently rub it in to the fretboard and over the Frets themselves. Leave it a few minutes and then give it a quick buffing with another clean rag. You'll probably find that the cleaning rag will be filthy once you've done this. Have a look at the finger board, if you thing it could do with another going over, do so.
Really rub hard when you are taking the oil off on the second application, your fingerboard should now be looking far nicer and your frets should be gleaming.
As far as the frets are concerned, perhaps pop a picture up on here so that we can get a better idea of how bad they are? It's a long time since I had any fret work done, so I am out of touch with prices, but a good fret dress WILL cost more than you paid for the guitar, but in my view, it would be money well spent.
Where abouts are you based? Might be able to make some recommendations on people woth using once we know your location.
String Breakages. 99% of the time the problem will lie at the Nut. But Showmasters came in many types, some had a Fender version of a Floyd Rose Trem and a Locking Nut, others had the more conventional style of Nut, so again, tell us which you have and then........ we can offer more advice.
What ever, the problem shouldn't be anything too serious.