Speed is no different from any other performance metric in cycling. Everything about it takes time and repeated efforts. It's all about physical conditioning.

Don't sweat that you're not able to sustain 15 mph right now. As you continue to ride, push your own limits a bit each time, and you'll see it improve.

And yes, tires can help. Skinnier tires generally require less effort to roll down the road. If you never take your bike off-road, there's little to be gained in having standard mountain bike tires on it. With skinnier tires you'll either go faster for the same perceived effort, or ride your normal speed with less perceived effort. Just don't expect the tires to get you all the way to where you want to be, speed-wise. It still takes work on your part.

Oh, and street tires will make your bike sound a lot less like a 4X4 with studded snow tires on it, as you head down the road.