For sport - there are a number of ways to get in.
Press - If you happen to work for a press organisation, or can submit to an press organisation, they can organise access for you to attend an event. This is generally done on a per event basis. Get in contact with the local papers, and especially the sports editor.
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 person about gaining access. This may not always work, but a direct approach can be best.
League - A league body (FA, etc) can directly give permission to cover events. In general, this can be a bit more complicated. Showing your ability or a portfolio may be required.
Accreditation - For some sporting events, you need to register for accreditation with third party companies. Again, having a portfolio is a good step, so they can review your ability.
Organisations - Join a recognised Press/Photography organisation, such as AIPS, NUJ, etc. This will only really work if you already have a body of work behind you, and a portfolio.
Best of luck.