What you need to know is what sort of reader / tag you have. There are three main types.
- 125 KHz
- 13.5 MHz
- UHF tags with a frequency in the GHz range.
The simplest to work with is the 125KHz tags.
To tune an antenna you have to add a capacitor or two to it in parallel with the antenna, you can add them in parallel so they will add up. The capacitors have to be high voltage because you can get over 200V on a coil in resonance.
You measure the voltage across the coil with an oscilloscope and apply a signal to it using a signal generator. You change the signal generator’s frequency and when the voltage across the coil peaks to a maximum, that is the resonant frequency of the coil / antenna. You then change the capacitance across the coil and measure again, and repeat this until you get the right resonant frequency. I used to use a capacitor substitution box for the capacitor I was working with.
So it is an easy procedure if you have all the test gear.
The 13.5MHz are a bit more tricky because the reader transmits on one frequency and receives on one quarter of that frequency. And UHF tags need advanced test equipment that will work at those frequencies, these are very expensive.