You'll struggle to get any direct Psychology experience while at university, just because there's hundreds of graduates desperate for experience already - though it isn't impossible.
In my opinion, the best experience to get is working as a nursing assistant/support worker in the NHS. You can join the bank (i.e. work as and when you want), the pay is good, the hours flexible and you get a ton of hands on experience. You can also work for private companies as a support worker.
Shadowing qualified Psychologists, or other similar stuff, may not be as beneficial as you might think - often your access to information is very limited.
Things like befriending, Youth Offending Service, homeless shelters, youth groups, etc. are all good places for direct experience of working with people in need.
You can also see if any researchers in the department will let you type things into a computer for them - over time they may allow you to become more involved in research.