Fred R
It's a great idea to get into a syndicate (a group share) ASAP because if you fly any reasonable number of hours, it will make flying a lot cheaper than self fly hire.
The plane must be on a Public (not Private) Transport CofA if it is to be used for any training for the initial award of a License or a Rating. Be careful with this since a share will be happily offered for sale to you without anyone telling you this (often shares become available because the group is trying hard to get shot of a troublesome member) and then you find you cannot do the IMC Rating in the plane, for example...
Insurance problems will depend on the plane but I found that the insurer wants just a valid type conversion (actually it is called "differences training" unless you are converting to a turboprop or a jet, in general) and you are likely to have say 10 hours on the type by the time you have finished doing that. There is generally no discount until you reach some 500 hours...