With regards to private tutoring, what would be the best way to find a good tutor?
Word of mouth. Ask other parents whether there is someone they know and could recommend and ask why - what in particular did they/their child like about the tutor in question? If you are going to have 1-2-1 tutoring, it's not only the tutor's knowledge and reputation, but also personality and teaching style that need to match what your child needs. E.g. some tutors are very strict/formal, some more relaxed in their approach and both can achieve equally good results.
If you have two or more names, talk to them on the phone to get a feel for their way of doing things. Do they have a standard approach they always use, or do they adjust it depending on the particular child's needs? Do they like parents to be involved, or do they prefer for them to stay out of it all? Do they set homework and if they do, how much?
I once heard of a tutor (first-hand account) who had her students work at the kitchen table while she was getting dinner ready. We didn't go for that one...