Press - If you happen to work for a press organisation, or can submit to an press organisation (newspaper, magazine, press agency), they can organise access for you to attend an event. This is generally done on a per event basis.
Portfolio - Your best bet is to build up some form of portfolio of images from sports events, to show you have the skill and talent to capture quality sports images. It can be a Catch 22 situation - you need images to get access and you need access to get the images. But, it is always worth your while practicing at local sports games. In general, your camera and kit lens is usually not enough to blag your way in to a professional event.
Contacts - Who you know can help. If you know someone in a local sports club then start with asking them. If you know someone in a club organisation, they can help too. Asking around is normally a good place to start, especially for lower league games.
Clubs - Sometimes you can just contact the club directly. Ask their administrators or even their PR/Media officer about gaining access. This may not always work.
League - A league body (IRFU, FAI, GAA) can directly give permission to cover events. In general, this can be a bit more complicated. Generally, this is required for professional organisations. Many of these organisations also require you to have full professional insurance, including Public Liability insurance. The amount of cover can vary from € 1 million to € 7 million in cover.
Accreditation - For some sporting events, you need to register for accreditation with third party media companies. Again, they may require you to have proof of insurance and also having a portfolio is a good step, so they can review your ability.
But, again, working for a press/media organisation is the most common way to gain access to big events.