It's been a while since i looked this up but;
Private catagory.
You can have instruction if the aircraft is wholly owned by the person (you)
In a group (you as one of the members) aircraft, the last 50/150 check must have been signed off by a licensed engineering place.
To use a random aircraft, not sure.
There has been a recent AIC on it and you can find the regs. in the ANO. It's all a bit confusing'ish/detailed and wordy, but i am told thats how the ANO was inetionally designed.
And to make life even more complicated. The regulations are dependant on the type of training the aircraft is doing.
Insurances a pretty specific to groups etc. so that is one for you and the FI to check with the operator concerned.
Your FI should be able to find out this plethora of legislation.