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There’s a funny line in Guys and Dolls where Sky Masterson, the suave gambler playboy, takes straight-laced Sarah Brown down to Havana. Her first drink ever is some combination of milk, sugar and rum and she quite likes it. “You know, this would be a great way to get children to drink milk!”
I vote Southern Comfort.
I second limoncello. It tastes exactly like lemon PEZ. Best made into an Italian soda. Midori also works. Chambourd tastes like raspberry jam with a hint of something flavorlessly-acidic underneath.
Alternatively, make your own. A friend and I developed the following recipe in college.
Acquire:
1 fifth of Cpt Morgan’s silver spiced rum
About a half pound of strawberries
Some variety of acidic fruit juice, or cheap lemonade powder
Pour out about a quarter of the rum. (Feel free to pour it into your own cup. It has to go somewhere.) Slice the strawberries into thin sticks or wedges and stuff them down the neck of the bottle into the remainder of the fifth. Let the bottle sit undisturbed for between twenty-four hours and forever. The strawberries will bleach out to a pinkish-white, and the rum will turn the color of strawberry ice cream topping. Use liberally to spike citrus juice or pink lemonade.
We used to make these with generic faux-Country Time pink lemonade mix, and refer to them as “pink things”. They don’t taste at all like alcohol, and pack a wallop. More than one friend who encountered them for the first time got much drunker than they intended, despite the warning. Other fruits (we’ve tried raspberries, black cherries, blueberries, blackberries, pineapple) work, but so far only strawberries make it turn really dramatic colors.
Have fun. 
I tried limoncello once, and only once. It tasted like turpentine mixed with Pledge and burning. I do not recommend giving it to someone with no liquor experience. Same for any of the anise flavored liquors already mentioned, and I LOVE anise. I have no idea why anyone would think ouzo is a beginner’s drink.
Although it’s very, very sweet it is very, very alcohol-y. I know because that’s what we girls drank back in high school. Yes, I have led a debauched life.
I think anyone could drink a bottle of coconut rum or peach Schnapps with little complaint.
Or is it just me?
Goldschlager. Shit’s 40% alcohol by volume (same as most whiskey) yet tastes like Big Red gum and not alcohol-y at all.
For a mixed drink that doesn’t taste like alcohol, order an Alabama Slammer. There are several variations but it tastes a lot like Hawaiian Punch fruit drink. And they are fairly strong.
They are so good that you may wake up looking at a stranger’s ceiling.
I recall a liquor I had a few months ago called Rumchata (I assume made to taste like Mexican horchata), that tastes pretty much like melted cinnamon roll frosting. I am not usually a fan of milky liquors, but man… it was sweet sugary slightly cinnamon goodness!
Tequila with triple sex…penis ensues. 
SoCo is usually 70 proof (35% abv), although there is a 100 proof version of it. I personally would not consider it very alcoholy-tasting. It’s only about 7 percentage points stronger than amaretto. I’d give it a shot, although I don’t like the stuff.
I had a Japanese plum wine a few weeks ago that tasted non-alcoholic, but I can’t recall the brand name.
Craz3d117 mentioned Goldschlager which made me remember a delicious not alcohol tasting thing that make sometimes. I call the apple pie milk-shake. Each serving is one shot of Goldschlager, twice that amount of apple cider (I suppose you could use hard cider but I never have), and half a scoop if ice cream. Throw all that in blender until it’s a uniform consistency, and sprinkle cinnamon on top to taste. Feel free to adjust the Goldschlager concentration depending on how drunk you want to get.
Proof doesn’t really have a lot to do with how alcohol-y something tastes, ime. There’s a particular flavor component that people are calling alcohol-y (or liquor-y in the OP) that’s like the smell when you open a bottle of rubbing alcohol, and the strength of that component varies wildly among drinks of roughly equal proof. I snork down a fair bit of bourbon and tequila, but find vodka (and most vodka-based drinks) undrinkable because it tastes so strongly of alcohol. I think it’s because the first two have a lot of other strong flavor components to compete with/mitigate the alcohol and vodka might at best have a little vanilla or fruit flavoring added in.
And yeah, to me SoCo totally has a stronger alcohol flavor than the nut liqueurs like amaretto.
That’s interesting. Like you, I find vodka “alcohol-y” tasting. I think I understand where the person mentioned in the OP is coming from. But I don’t particularly find SoCo that alcohol-y. Which is why I say “give it a shot” because, in my experience, I’ve found people that don’t like whisky, bourbon, vodka, rum, etc., often enough enjoy SoCo. In my circle, it was a total “chick drink” for that reason. But taste is obviously subjective and I don’t expect everyone to agree on whether it’s alcohol-y or not.
Thirding/fourthing margaritas. They never taste like anything but lime drinks to me. I can never manage more than a single glass of beer or liquor and rarely more than half a bottle of wine, but if I’m not careful I could drink margaritas all night.
Mango-flavored Malibu rum. It’s really kind of frighteningly close to mango juice.
Peppermint schnapps. It’s clearly very strong, but the peppermint thoroughly masks the alcohol taste. It’s very hard for me not to pull a face when drinking more typical liquors, but schnapps is just pleasant, especially chilled.
Barenjager is like drinking thinned honey. Personally I find it really kind of nauseating.
A properly made Swedish fish (Black Haus, Peach schnapps and a splash of cranberry juice) is easy and tastes pretty much like the candy.
If the friend likes sweets, butterscotch schnapps never tastes the least little bit liquory to me. If I’m cornered into accepting shots from someone, I’ll request butterscotch schnapps. If I’m cornered into accepting an actual cocktail, I’ll request a white russian, kahlua and cream, or amaretto and cream. I’m a serious lightweight, mostly because I HATE the liquory taste. Gin is right out…god, it tastes like isopropyl smells…
There are a lot of new flavored vodkas out now, and some of them have very little alcohol taste. I had one called Wedding Cake recently and, while sweet drinks aren’t really my thing, I could see how it would appeal to someone who doesn’t like the taste of alcohol. Smirnoff makes flavors like Fluffy Marshmallow and Whipped Cream.