Lots of good advice above. It's a balance based on personal preference, time, money, and space.
Smallest budget/space, don't do anything, ride your knobbies everywhere (bonus: exercise! bonus for some: noisy, vibrate-y rolling)
Next increment on the budget/space scale: buy replacement tires, swap them out as necessary.
Next: buy replacement wheels+tires, quicker to swap out, a little more space to store, more money to buy (but look for cheap MTB wheels on craigslist! you should be able to find decent wheels for $20-30 ea)
Next: buy a separate bike for the road: obviously this requires the most space & money.
Maybe the best strategy is to start with spare tires, try that for a while, and if you want more convenience, get some spare wheels as well (and you can still use the spare tires you already bought). In the meantime, it is a Very Good Thing to get practice at changing tires. If it's a big deal for you to change a tire, then it will totally ruin your day to get a flat on a ride, and might put you off cycling for good (thus jeopardizing your fitness goals). But if you are well practiced at taking tires off and putting them back on, a flat on a ride will be merely a 10-15min inconvenience. If you know you can change a flat on the road, you will have more confidence to take longer rides, further from home.
My personal recommendation: buy a pair of Nashbar Slick Wire Bead Mountain Tires. On sale for only $10 ea at the moment but usually can be had for $12; that's virtually free in the world of bicycle tires. If you check the user comments at Nashbar, they are very well reviewed for ride and durability. I have recommended them to friends and they were very happy with them too.
My personal story is that I didn't ride bikes for about 15 years after I finished school, bought a cheap MTB off CL, and it was just "Meh." When one of the tires exploded (it was an old tire on an old bike from CL, not surprising), I ordered a pair of these slicks and WOW, cycling was fun again. Maybe cyccommute is right and road tires are only a few mph faster, but I sure FELT a LOT faster, and I was also rolling smooth and quiet (if you enjoy rolling noisy and vibratey, then stick with mountain knobbies!). I credit these tires with restoring the joy of cycling for me, and catapulting me back into the world of cycling.