WARNING: This circuit can produce LETHAL voltages and is provided for information only.
You cannot just use any old transformer. A salvaged fluorescent tube transformer is very unlikely to work.
If you don't (or can't) wind your own transformer then at least go for a transformer you know is designed for the job.
(1) Car ignition coil
(2) A line (flyback) transformer from an old CRT display
Both are designed and made to generate very high voltages
Things you should notice in this circuit.
(1) The 555 is decoupled from the 12V supply through a 56 ohm resistor (R3) and 100uF + 0.1uF (C1,C2). Try to get these capacitors connected as close to the pins of the 555 as possible (pins 1 and 8).
(2) The circuit does not require the 2N2222 transistor to drive the Mosfet. (Neither does your circuit). The 555 is quite capable of driving the gate of a Mosfet which is essentially a small capacitor that needs a sufficiently large voltage change (0 -12V is fine) and not a lot of current to turn it on and off.
(3) It uses diodes (D3,D4 - rated at 600V,3A)) to protect the Mosfet from voltage spikes and reverse voltage.
(4) It uses a different Mosfet - IRF640. The main differences are a higher maximum voltage and a lower turn on resistance. The lower turn on resistance means LESS POWER (I^R) will be lost in the Mosfet and it will run cooler (although you still mount it with a heat sink).
A suitable transformer should be available online for shipping to you.
What you propose is a bit risky as under light load the voltage may rise higher than you hope.
However depending on the gate circuitry it may be tolerant of 24V at some increase in operating speed and decrease in motor and/or gearbox lifetime.
Gate circuitry might possibly accept DC in which case you can add in a diode bridge, smoothing capacitor and a regulator to your transformer to get the ideal voltage.
An alternative might be to find a suitable size (bigger than what you think you need) transformer with multiple taps and use it as an autotransformer to adjust your voltage to suit. You can use taps designed for higher voltages as long as the current rating is adequate. Say 48V and 36V taps would work to bring 24V to 18V as well.
Best Answer
1 minute on google found this: -
This will be about 70% efficient and probably a bit more efficient than using a 555. Read the data sheet on the LT3573 for detail on what transformer they recommend.
Then there is this beast: -
I used the term "5V to 300V converter" to search for images and note that I changed 6V to 5V because it is more likely to lead to a better search result. You'll also need to be able to generate 300V as the batteries droop a bit too.