The fun thing about single malts is that they showcase more of the distillery's personality. You can taste some really wild whiskies from places like Islay, or stay with the softer, malty whiskies in Speyside. Laphroaig would be a good example of the former, while Glenfiddich, Macallan or Glenfarclas would be a good example of the latter.
After the whisky has distilled and matured, some whiskies are then "finished" in port casks, sherry casks, or madeira casks, etc... It is fun to taste the differences these finishes can give a whisky. Glenmorangie has whiskies in several different finishes that makes this a fun experiment: La Santa, Quinta Ruban, Nectar D'Or, Sonnalta PX, etc...
Additionally, some whisky is finished in different kinds of wood casks (French oak, new oak, etc...). The wood can give the whisky a unique flavor as well. The Glenlivet 15yr French Oak is a great bottle of whisky to display this characteristic.
Other than how it is made, whisky is also bottled differently. The Balvenie 15yr Single Barrel is bottled straight from one cask, as opposed to most single malts which are a product of different casks from the same distillery being "vatted" together in search of a specific flavor profile. This gives you a "raw" tasting of the whisky straight out of the cask.
Other than being a single cask offering, some whisky is bottled at cask strength, instead of being watered down to 40-46% ABV. Cask Strength whiskies are in the high 50's to low 60's! My favorite cask strengths are Macallan Cask Strength, Aberlour A'Bunadh (which are very similar whiskies) and Laphroaig 10yr Cask Strength.
What I would recommend doing is to go in search for sample packs. I have seen sample packs of The Balvenie (12yr Doublewood, 15yr Single Barrel and 21yr Portwood), and Glenfiddich (12yr, 15yr Solera Reserve and 18yr.) Then find as many 50cl bottles as you can find of the various single malts. I have had good luck finding various offerings from Glenlivet. I even found a small 50cl bottle of Macallan 18yr Sherry Finish, which is quite excellent.
After you have tried as many small samples as you can find, and after you have tried everything the local taverns have to offer, then come back with a ranking of which ones you liked the most. That will help us steer you in a direction you might like to explore first!