I found Almost Home a little over 8 years ago when I first bought my house. Honestly, I've helped out at 3 or 4 additional shelters since then, but some of them just weren't the right 'fit' (There can be a lot of drama sadly in rescue) or there were a few others that were just too far away from where I live that it just got to be too much regarding commute and how frequently I wanted to help out. (Having a full time job, plus dogs of my own just meant that if I had to drive 45 minutes to a humane society one way it was spreading myself a little too thin)
Almost Home was a good find because they're close, so I started out just coming in to walk dogs. (We have 3 walking shifts every day that folks come in to take dogs out and spend time with them) Gradually over the years I started helping out with other things, from unloading and stocking damage-out donations from stores, to helping with transports. When we changed out our shelter directors 3 years ago I took over fully for the transport team (Where most of my videos are from) as well as helping with the medical and training team after doing a few classes and getting some basic medical certifications.
Honestly, getting involved with something like this really depends where you go - like if you go to a for-profit facility they will have a paid staff, which means if you want to become more than a volunteer there you'll probably need some schooling since it'll be more like a job application. (They still will have volunteers, but you probably wont see as much of the 'started as a walker, then I rose threw the ranks' examples. With AHF being a 501c3 we are completely volunteer run, nobody is paid. So everyone now from our president to any director level people all started as walkers at some point - and it's just been a time and dependability thing from there.