I think it is a good idea to go private if you can afford it.

People typically say that the best way to find someone is to get a referral. Do you have a doctor you can trust, or someone else you can trust to give you a referral?

Otherwise...

An option could be to have a bit of a look online for what kind of therapy you would like. CBT, psychodynamic, etc. That will help you narrow the list down a little.

What are you wanting to see a therapist for (what kind of issue?)

I've heard people (who didn't have a referral) would just phone people fairly much at random. Or phone people who were close to where they lived or wherever.

I think it is wise to not have a public record if at all possible. A therapist should be able to figure whether you would benefit from a psychiatric consult for medications too. I think it can be a good idea to try therapy before trying medications because there can be lots of side effects etc associated with medications.

Good luck.