Experience is key, even for junior paid roles. Volunteering is a good place to start. You will find some useful information and links in the Building Experience section. Think about what areas you are most interested in. If you want to get into research or advocacy/lobbying you may be looking for different experience to on-the-ground programmes and relief work.
However, be flexible. It can be useful to get into an organisation and build from there, so don't discount areas you haven't thought about yet...
If you have extensive voluntary experience and/or a postgraduate degree, start looking for internships in NGOs and international organisations. You may want to look for entry-level paid roles, too.
Many graduates aspire to work, or get internships, at the UN
There are limited paid opportunities and the processes will be highly selective
There are internships throughout the UN system, but they are unpaid and extremely competitive. International field (and related work) experience, plus language skills and a Master's degree are often required