I have an old, rather low-quality 18-speed mountain bike that's been sitting around for a long time, and I think it would make a fun project to buy some of the parts and convert it to a road bike (with the thin tires and rims, and curved handle bars, etc).

I think the frame is in decent shape, and maybe the fork (no front suspension, just metal), but since i'd replace the wheels anyway, the fact that they're fine doesn't matter. The derraileurs are probably junk, even if they do work, but I think the hubs (is that what they're called, the things with sprockets the chain rides on) are OK.

How much would it cost to replace just the wheels and tires, handlebars, and probably chain, derraileurs and shift levers?

Of course I don't need top-of-the-line componentry for this bike, and I'd still ride my mountain bike most of the time, so quality/toughness isn't the biggest issue.

It probably isn't worth the cost. After replacing the wheelset and componentry package you could probably buy a decent entry-level road bike. Or a used road bike. The wheelset will cost you over $100 most likely. The STI-shifters cost $120 at the least. You're already up to over $220 probably quite a bit more for the wheelset, and you still have handlbars which will be $80+ at least, and derailers to buy. You could probably get a used road bike for as much or even less. I know that the shop I go to had a lightly used Schwinn Sport for $399. If you really want to do it just for fun, try looking on ebay for the components you need. You might be able to find some deals on there. You could check garage sales, etc for old bikes that you can strip down. If you go for normal brakes with barend shifters you might be able to save a little money.

It'll cost more than you think. The cheapest way to buy bicycle parts is already bolted onto a frame. You might be able to find a donor bike to salvage the parts that you need off of. If you do, however, you might want to reconsider which of the frames you want to build up.

don't do anything more than get skinny tires for it.

You can get one inch tires for 26 inch from Specialized now, which are the thinnest I've seen.
The popular catalogs have 1.125 inch for around 10 bucks each.

For an old junker like that you can easily turn it into a for almost no cost at all.

Thanks for all the comments and suggestions. I might check yard sales or ebay for some thin tires and rims, but I actually might be better off putting them on another bike I have, it's in better condition, but needs tires.

*singlespeed