I'd suggest the Ardbeg 10. If you were going with an older Laphroaig or the Quarter Cask, I'd say Laphroaig is easier for an Islay-newbie than Ardbeg. But I find the Laphroaig 10 to be rough around the edges. In contrast, although the Ardbeg is heavily peated, you'll also get some fruit and other flavors. And the Ardbeg 10 is more refined than is the Laphroaig 10. All IMO - maybe others would disagree.
Also, I'm sure others will add that neither Laphroaig nor Ardbeg is the gentlest introduction to Islay. Something more like a Bowmore or Lagavulin might be a good place to start (not counting the not peaty Islays).
I'd suggest the Ardbeg 10. If you were going with an older Laphroaig or the Quarter Cask, I'd say Laphroaig is easier for an Islay-newbie than Ardbeg. But I find the Laphroaig 10 to be rough around the edges. In contrast, although the Ardbeg is heavily peated, you'll also get some fruit and other...