How long does it usually take to get an appointment with a new therapist? Are there any walk in clinics that are also cheap/free for those without insurance? How does one go about finding their local resources?
It depends a lot. There are walk in clinics that can see people immediately, and in some clinics have special walk-in days or times, or clinicians who dedicate a piece of their time to seeing crisis or walk-in clients. usually public mental health programs has a long waiting list, often in the months range, so it can be hard to get into those programs. there are places like the VA which have a mandate to see clients essentially immediately after they express interest or need, so the wait is more like minutes or hours and maybe a day.
I’d suggest looking at NAMI or calling the NAMI Helpline if you’re in the US- they have local affiliates who can identify programs and therapists. You could also look at the Psychology Today listings, look up your state/county/city health and human services department, or call local universities, who often have psychology, psychiatry, and/or counseling clinics for community members.