Unless you're a fully-fledged race team, filling your tyres with nitrogen is a comprehensive waste of money.
Tyre retailers offer nitrogen fill-ups at between $5 and $10 per tyre - so, between $25 and $50 per car. (Most cars have five tyres, counting the spare.)
Filling tyres with nitrogen is a classic automotive 'upsell' - the tyre retailer can get you in with a low price on the tyres, and sell you hugely profitable nitrogen re-fills ... which are about as useful, in the real world, as an ashtray on a Harley-Davidson.
This practice is borderline unethical and certainly a waste of your hard-earned cash.
The purported benefits of nitrogen, claimed by tyre retailers, are:
REDUCES TYRE OPERATING TEMPERATURE
While this is true in a physics-lab sense, the absolute amount of reduction is minimal. This is an example of hyping up the truth to the extent that is might as well be an outright lie. The benefit to average motorists is zero - if you're a Formula One race team, it might be good for 0.01 seconds a lap. On a good day.
IMPROVES RIDE QUALITY
There's no scientific reason why this could be the case. Compressed air and compressed nitrogen deliver exactly the same ride quality. Claims about improved ride quality are bullshit - pure and simple.
INCREASES TYRE LIFE
This also is unmitigated bullshit - what increases tyre life is: Regular pressure checks and top-ups when required, driving more sedately, rotating the tyres to different wheels when the car is serviced, and checking for uneven wear, which is often a sign of poor wheel alignment.
REDUCES PRESSURE BUILD-UP
This is a flipside of the reduced operating temperature argument. Obviously, when a tyre gets hotter the pressure increases. (This is why you should check the pressures cold.) It's immeasurable.
SLOWS DOWN THE RATE OF PRESSURE LOSS
This is true - nitrogen molecules are bigger than oxygen molecules, so they don't leach out through the rubber as fast. But the difference in the rate of pressure loss is, again, minimal. And the downside of this claim is that many owners simply think "I've got nitrogen; I can afford not to check the pressures regularly." This leads to increased wear and less safe operation.
WON'T CORRODE THE WHEELS
True - nitrogen is 'dry' or 'anhydrous', which just means it's supplied with zero humidity. Hypothetically, that makes it harder for the wheels to corrode. However, this is again a grossly overstated claim by the retailers - when was the last time one of your wheels rusted through from the inside?
AIRCRAFT TYRES USE NITROGEN
True - but only some aircraft, typically high-altitude jets. They do it because nitrogen is inert, and using it reduces the risk of explosion at 30,000 feet (or somesuch) which could theoretically destroy the aircraft. Not much of a worry on your Falcon or Commodore. (Unless you're planning on driving to the summit of Everest...)
Nitrogen is sometimes used in vehicles driven in mines and other hazardous areas - because nitrogen doesn't support combustion.
Nitrogen is expensive, the benefits are grossly over-stated, and it's not available everywhere - so if you top up the pressures at your local servo, you're introducing regular air into the tyres anyway.
Why not do this: take a big, deep breath. If your lungs are in good nick, they hold about five litres of air. Think about the volume of air filling up your chest. Four of those five litres is nitrogen. (Atmospheric air is about 80 per cent nitrogen ... it's actually 78 per cent nitrogen, 21 per cent oxygen and one per cent a cocktail of other gasses.) So, when you fill up with free compressed air at the servo you're getting 80 per cent of the purported benefits of a nitrogen re-fill ... at a saving of 100 per cent.
The decision to decline the 'offer' to fill your new tyres with nitrogen should be a 'no brainer' - which is approximately the intelligence you're demonstrating if you agree and pay the $50 for something that yields no tangible benefit.