Due to the nature of the aux belt – in that it's a relatively simple toothed strip of reinforced rubber – the only two problems you're likely to face are either the belt slipping, or the belt snapping!
In the event of a slipping belt you may find the components it's powering struggling to work – intermittent power steering, flickering lights etc, and you'll pretty much always hear the belt squealing particularly when you first start the car from cold.
The embarrassingly loud nose will increasing in pitch as the revs rise. If your belt starts screeching don't ignore it. It may just need the tension adjusting or it may need to be replaced.
As the belt is pretty much always visible within the engine bay (and possibly also through the wheel arch) keep an eye on its condition – even very worn looking belts can work without apparent issue – but a damaged or worn belt should be replaced straight away – failure causes critical loss of all the devices the belt powers – so no power steering, no water being pumped round the engine, no charge going to the battery!
Many belts can be replaced with a basic tool kit. Removal is a matter of releasing the tension applied to the belt (often via the tensioner, sometimes via the alternator mounted on a swinging bracket) and sliding off the belt.
However, cars with tensioners may need a special tool to release the tension. There are several different tools depending on the application. They are commonly a long thin bar with some kind of socket that connects to the tensioner.
Tuning the bar releases the tension allowing you to remove, then replace the belt. Due to many belts having limited access, even if you only need to slacken one nut you may need to invest in a long handled spanner, which will not only give you access, but will also increase the leverage, as they are seldom easy to undo.
As with all serviceable parts on a car, you should be prepared to replace it at the scheduled service intervals, or whenever there is an issue with it.
Keep the engine bay clean but don't be tempted to apply any lubricant to the belt which can cause it to slip.
How long an auxiliary belt lasts depends on how much use the car gets, and the recommended service intervals.
It's not unusual to get 50,000 miles from a belt, but at the same time don't be surprised if yours lasts a lot less time – particularly if it becomes contaminated with oil or similar.
You should check the belt regularly for signs of wear, damage, or cracking and change it at the first sign of degradation. A very shiny belt is also a sign replacement is potentially overdue.
In general the cost of the actual belt isn't too great – usually between £10 and £20. If a special tool is required to facilitate the loosening of the tensioner, then you must also factor that cost in.
Expect to pay £10-30 for the correct tool if yours can't be removed with a regular tool kit.