Is there anything I can do to help minimise the scar from my shave biopsy?

Shave biopsy/removal of a mole is an excellent technique and when done correctly, can yield excellent cosmetic results. As the sites are healing, I would emolliate them with something bland and greasy like Aquaphor. I would also protect them from the sun if they are being exposed to the sun.  If they start to look like they are going to thicken up, follow up with your doctor who can do intralesional cortisone injections. You may also want to try a silicone based scar gel with sunscreen. 

Shave excisions remove the upper most layer of skin and the chance of scar is minimal.   However, some people genetically make thicker scars or keloids.  Moles removed from the mid chest and back to the neck have an increase risk of scarring due to location and constant movement.  I like the use of petroleum jelly as the surface heals.  The use of tegaderm patches applied for approximately 3 months as the area remodels is shown to be beneficial at reducing the appearance of scar.

Remember that it may take 6 months to a year for a wound to heal on the torso.  I wouldn't recommend any aggressive treatments such as a laser to remove moles.  Pigmented lesions should be biopsied, and lasers would only burn them possibly causing increase risk of scar.

I hope this helps!