It's very difficult to know exactly what to buy for a second lens, because everything sounds useful and better. And of course there's massive sticker shock when you find out what they want for some lenses.
A lot of people want more telephoto range beyond the kit lens. Two popular budget choices are these:
Tamron | 70-300mm f/4-5.6 Di LD Macro Lens | AF017P700 | B&H Sigma | 70-300mm f/4-5.6 DG Macro Autofocus Lens | 509109 | B&H
Both of these are autofocus, with very similar specs. They were built to be the second lens choice, and they do a decent job.
Pentax has some lenses for the same purpose. The old choice was the DA 50-200. This lens has two other variations, the DA-L with a plastic mount (possibly only available used) and the WR (weather resistant) version. They recently released the DA 55-300 to compete with the above Tamron and Sigma lenses. The Pentax is better but it's around $350.
Another popular path is to get one lens to cover every possible need. I tried this myself, but be aware that the cheaper lenses in this category will compromise image quality. Pentax had the DA 18-250 for a while, but stopped making it. It's harder to find now, and may be out of your budget at $450. Expensive, but everyone likes its image quality.
All of the above lenses will only open to f3.5 or f4, and then only at the wide-angle end. Some people find that really limiting, so they look for "faster" lenses that open to f2.8. Ash mentions some good choices above. The only way you can get faster lenses on your budget is to look at used prime lenses, often manual focus ones. That's what I did, but it's not for everyone.
Another direction you could take is to get better wide-angle coverage. I don't think it's worth pursuing this on your budget. The kit lens is pretty good at this end. Better lenses are out there but costly. A used DA 16-45 for $250 will be an improvement, but not as big a step up as you could get in other areas.
For some reason, getting a flash is always a low priority with beginners. There are a few possibilities within your budget, like the Pentax AF360FGZ or Metz 48. Although flash requires some technique to get decent results, it can reduce your need for a super-fast lens and is much cheaper. My brother has your exact camera, and he found the flash way more useful than all of the lenses I offered him. He even uses his camera professionally, producing web content for Tribune Media.