I used to work as a volunteer for a charity called Hope House, they offer respite care centres for terminally ill children. I was basically just an assistant in one of their small shops, which is how the raise a lot of their funds. I just gave up a few hours on a Saturday morning, restocked the store, cleaned, operated the till. Everyone was really lovely there, it was a very good experience.
I also temporarily did a little bit with the Cats Protection League - cleaning the animals pens before the public could come in for viewing and rehoming. I also used to do what they called socialisation which is a fancy term for playing games and cuddles!
Currently I'm helping out with Samaritan's Purse and their Operation Christmas Child project. I became a registered volunteer a few weeks back and since then have been at one of their warehouses a couple of evenings a week. All the shoeboxes of gifts have to be checked and sealed before dispatch, so that's what I do. Involves going through the boxes, taking out anything unsuitable or filling up boxes which are slightly empty, then taping them up, sorting them and sending them on their way. Again all the other volunteers I've met have been really lovely, there is a nice atmospehere despite the warehouse being absolutely freezing ha ha!
How I got into it was through the internet. If you know of a charity or organisation in your area it's a good idea to google them and see if they say anything on their website about volunteers or if they have any contact numbers that you could get in touch by. It's good to actually speak to the person in charge and leave a good first impression, before you turn up to meet them. Most places are usually more than grateful for any time or help you can provide though.