A blister occurs when friction and other stress on the skin causes the superficial levels of the skin to separate from the deeper layers, with liquid accumulating between the separated layers. A blister is evidence that the stress that you are putting on the skin is more than it is accustomed to or can handle. Blisters can be minimized by having good shoes and not over training; however, if you are just starting to run seriously, probably a few blisters are unavoidable.
Your friend is technically wrong that blisters turn into callouses. However, the idea is right - as your skin is subjected to the stress of running, it will start to grow thicker and tougher - forming areas of thick protection (callouses) over pressure points and other sensitive spots. These are not blisters "turning into callouses" but they do represent the areas of the skin where the blisters are forming now toughening up and becoming blister proof.
In the meantime, you may want to check with a shoe store to make sure that your shoes fit and are not the cause of the blisters. You can also try wearing moleskin over the areas that are blister prone. And, of course, talk to your primary care doctor
for more specific information!