If by "reading glasses" you mean the glasses you can buy off the shelf in UK chemists, supermarkets, etc, with no eye test, don't go down that road.
Get a proper eye test, and explain that you don't want "reading glasses" that only work at a short distance from your eyes but you need to focus on longer distances. Measure the typical distance from your eyes to the music stand and the keyboard, so you (and the optician) aren't guessing!
FWIW, I made the mistake of asking an optician for "reading glasses" (because I didn't realize that was the wrong description of what I wanted) when I started using them a couple of years ago. The first pair were useless for anything further than about 2 feet from my eyes. After explaining what I really wanted (i.e. glasses that will focus without any apparent magnification up to about arm's length away from me, for computer work) the prescription was altered and I got a free replacement lenses.
I don't know what the system is in Australia but in the UK, NHS-standard eye tests are either quite cheap or completely free(*), depending on your age, even when done by private opticians - you don't need to wait for a medical appointment at a hospital clinic or whatever, and you don't need to see your doctor before making an appointment for the eye test. Once you have the prescription, you can buy the glasses from anywhere you want, including by mail order - you don't have to pay the opticians' inflated prices unless you really like the style of the glasses they sell.
You can also get glasses with varifocal lenses which will give you a different range of "in-focus" vision in different directions - e.g. for looking at your hands and also reading the music - though personally, when I tried them out I found there was no extra benefit to justify the higher price.
(*) There are a few restrictions - e.g. you can only get a free re-test every 2 years, unless you have been diagnosed with a disease that may potentially damage your eyesight, such as diabetes. Note, the NHS eye test and examination should pick up early warning signs of such conditions even if you don't have any other obvious symptoms - IMO that's worth having as an "insurance policy" quite apart from fixing your current vision problems!