Agreed that the Laphroigs, Lagavulins (as well as the Ardbegs) are all most likely way too peaty/medicinal for a starting point. And it is true that the Scotch whiskies are overpriced.. it's largely a current upswing in popularity, but I've yet to be able to find something to truly replace it in my cupboard
Try a couple lightly peated malts like Talisker 10 or Highland Park 12. They're both very nice and won't break the bank. Oban as mentioned is really good, but a bit pricier, if I recall.
I'm not sure I'd send someone running for Bourbon as the fact that it's finished in 100% new
American oak means it's pretty loaded with vanilla and other oak traits. Know what you're buying. If you're ok with oak, Bourbon is great. Try Buffalo Trace, it's very good and pretty cheap relative to Scotch.
I don't have a lot of experience with Irish whiskey, but I can recommend Tyrconnell.