Yeah, you could put skinnier tires on there, but it's going to hurt your ride quality. I would never go below 28c on an upright handlebar bike - you get very marginal (if at all) speed improvements below that point but at a large decrease in ride quality. Even so, a bike that comes with 38c tires is likely to still be a crappy ride at smaller tires sizes.
Frankly, at least one other commenter is right - if you wanted a road bike you should sell this one and go out and buy a road bike. A decent entry level road bike costs around $800 retail, but if you can find one on craigslist or ebay or a local used bike sale it can be half that. Heck, there are local shops around me that sell old (70's and 80's) road bikes they've fixed up for $100-$300 that would make a better road bike than this one - I don't even *recognize* the front or rear derailler names, they're that low end.