I genuinely thought that Chickenpox is typically a mild illness in healthy children,
Typically, yes, it's "mild", if you consider a week of discomfort and common scarring for life "mild". Every now and then it kills a child or hospitalises them at random. Not a chance I'd take.
and contracting it naturally often provides lifelong immunity
As does the vaccine, without having to go through the illness.
and also vaccinating against it could potentially delay the infection to adulthood?
Evidence shows that the chicken pox vaccine confers immunity for life, provided you have both doses in the recommended timescales.
when the disease is more severe and my dr friends have always told me natural chickenpox infection might reduce the long-term risk of shingles by boosting immunity.
Studies have shown there is no increased risk of shingles and in fact, limited evidence appears to show a decrease in the chances of shingles as the viral load of the vaccine is far less than the illness.
Any side effects of the vaccine ?
All vaccines and medications have possible side effects and I'd recommend you read the literature from your vaccine manufacturer. In my daughter's case, there were no side effects and she didn't even notice the needle go in as it was so quick and painless.