Tyrconnell is a pretty darn good single malt. It is a twice-distilled, unpeated Irish whiskey, and is a good entry into single malts. Another you might try if you want to stick to Irish single malts is Knappogue Castle.
The world of single malt Scotch whisky is much more broad (Ireland has four active distilleries, compared to some 90 in Scotland). As I noted in another thread, some of the more approachable single malts from Scotland include Glenmorangie Original, Glenfiddich 12, and Auchentoshan 12. If you find you enjoy the Irish whiskies in particular, I would add some weight to the recommendation of Auchentoshan as it is unpeated and triple-distilled. If you want to kick it up a notch, you might also consider Springbank 10, Macallan 12, or Highland Park 12. These are toward the richer end of the spectrum, but remain rather approachable for someone starting to explore single malts, and are renowned by single malt Scotch whisky enthusiasts while not breaking the bank.