I would never, repeat never, offer an Islay, let alone Laphroig, to a new Scotch drinker. That's like giving Guinness to new beer drinker.
Laphroig is the most extreme example of a peaty, medicinal, Islay. It is no wonder that new Scotch drinkers would find it difficult to appreciate.
The 10 y.o. is the most medicinal and peaty. To understand the Quarter Cask you have to go back to the old days when the whiskey was transported by horse and dray. The large vats were too big to be hauled over hill and dale so they used smaller casks about a quarter the size of the vats, hence the name. The whiskey would slosh these casks as it was transported and this contact with the oak brought a richer, mellow character to the whiskey, a "sweeter" note if you will.
To understand this concept of sweet in whiskey think of people who are drinking beer for the first time. To them all beer is bitter so they would not be able to differentiate between beer that is hoppy and those that are "sweet".
Anyway, I wish you all the best in your journey throught the world of single malt whiskeys.