Originally Posted by RonH
The cheapest and easiest thing you can do is replace the tires. 700x25 or maybe 700x23 if they'll fit the rims?
I've actually done this a couple of times--I got back into cycling in the '80s with a mountain bike after a 10-year post-college layoff, then "roadified" that bike and another so I could do road rides until I could afford a roadie. I even did a couple of centuries on a mid-'80s, 30-pound Mongoose mountain bike with 1.5-inch road tires.
Tires are by far the best bang for the buck. I wouldn't go to 23s unless you're a lightweight (under 150), but 25s or 28s will fit, and you can run them at 95-105psi. You'll notice an immediate difference. you may need new tubes to go with them, but that's a minor expense.
It doesn't make sense to start swapping things like shifters, and your gearing is very close to what I use on my Atlantis and Rambouillet (46-36-26 triples with 13-28 cassettes). That's perfectly adequate for the road for most riders. Don't even think about changing the gearing unless you're consistently spinning out (pedaling as fast as your legs will move) in your highest gear--if you have high gears you don't use now, there's no point in changing to even higher ones.
Position is important, as another post mentioned, but maybe not supremely important at the speeds most of us ride. If you cruise at 20mph, you might pick up to 21 or 22 by getting into a classic roadie position, but you'll sacrifice comfort and visibility, and you'll have to buy new bars, shifters and probably a stem. If you cruise at 14 now, the swap would maybe bring you to 14.5 or something. Only you can decide if it's worth the money, time and inconvenience. A lower, longer stem ($15-$40 if you shop around) would lower the existing bars and move them farther away at much less cost, if that's what you need, but don't go too far at once. You won't need new shifters or anything else if you do that.
A lightweight wheelset would be nice, but you're probably talking $300 or more to upgrade a fairly low-end ride. I'd swap the tires, maybe the stem, and ride it like that until I figured out what I really needed. Keep the old parts, and when you decide to buy a real road bike, you can convert this one back as a commuter, beater or loaner.