Welcome to the boards.
I agree, it can be a mystical proposition, and even the pros screw tasting windows up. Basically, it all comes down to balance. Someone else can chime in about white and sweet wines, but a good ageworthy red has nicely balanced fruit and acidity and tannin structure (that astringent quality in some wines) and alcohol. Not enough (or too much) of any, and the wine won't age well. Determining what's what requires lots of experience, much more than I have, to be able to do with modest accuracy.
Here's an easy experiment, head down to the best wine store in your area and pick up an $8 and a $25 Sangiovese based wine. A couple I'd recommend, and are fairly common: Di Majo Norante Sangiovese, and a Frescobaldi Nipozzano Chianti, from same vintages, or close to it. I like both wines, and both will have good fruit (you'll probably smell and taste cherries, along with some other things like oakiness and maybe some plum). But, the Nipozzano will be more tannic, a bit more "intense." Neither are built for the long haul, but the DiMajo N. is built to be enjoyed sooner, and you'll probably find it less complex, sweeter, but perhaps goes down "easier" now. But, give that Nippozano two or three years, and it'll be similar to the DiMajo Norante, but with the addition of secondary characteristics that wine lovers go ga-ga over. Stuff like chocolate, subtle spice, maybe some notes of tropical fruit and flowers.
If you want to spend some more cash, get ahold of a younger Brunello di Montalcino, 1999 or 2001 is fine (avoid 2002), and you'll really taste a wine that has good fruit, acidity, but with overpowering tannins that will beg out for time in the bottle. Let it sit in the glass, or a decanter if you have one (for several hours, or overnight), and you'll notice it soften markedly. This somewhat mimics (but does not duplicate) the ageing process. If the fruit, alcohol, and acidity isn't there, that Brunello will be crap in a few years, and you'll end up with something that tastes like an old catcher's mitt.