I couldn't agree more with everyone's suggestion to start by volunteering. I'm currently on this journey and here are a few of my lessons learned:
1. Volunteer with an existing animal welfare organization, especially one with clear processes and protocols, structured volunteer training and clear opportunities to gain hands on experience. I learned the hard way by volunteering with small rescues that IMO bit off more than they could chew and reinventing many wheels along the way which resulted in a lot of frustration and missteps for me. Volunteering with a small rescue can have its advantages as well - seeing the problems experienced and looking for solutions but in my volunteer work and professional work, I am of the opinion that I'd rather learn the foundation knowledge first and tweak to custom fit later. Burn out / compassion fatigue is a real problem in animal welfare and anything that can be done to reduce the stress, the better.
2. Keep learning. As you may already know, there are many great learning resources (books, webinars, conferences, etc). The more you learn about the many facets of animal welfare work through these resources, the better you can apply them as a volunteer. You'll slowly start to find yourself speaking the lingo.
3. Keep your eyes peeled for job openings with large organizations like the ASPCA or HSUS. More often than not, they post professional positions that aren't hands on with animals but are important functional areas of any organization to keep afloat.
Best of luck!