I want to give the smallest practical amount of BTC to friends and family to introduce them to it.
this is a very good idea, just make sure that you know everything yourself so you can answer their questions easily.
What is the best way to do this?
there are a lot of ways, most popular one is to give away a paper wallet you made yourself. especially with a nice print and a short help on the back of it about how to spend the coins by sweeping the private key you give them.
What are the smallest amounts that can be sensibly transferred?
Thank you.
the smallest amount (if i am not mistaken) is the dust amount which is around 5600 satoshi.
but you should give something away that has a real value like at least $1 anything lower than that is not even worth giving since they can not really do anything with it.