I'll base my suggestions on a budget limit.

If you want to take more photo's inside of the house, the kit lens might be a bit to slow. I myself experience this and I've found that the AF-S 35mm 1.8f/1.4f is a very good way to avoid using a flash. The popup flash is not bad, but the results are very flashy, another way to solve this, is to diffuse the light, a nice working DIY solution for this is: DIY diffuser using a old film canister. I have used this and am very pleased with the results.

For trips I would say, get the AF-S 55mm-200mm VR, a very nice addition to your kitlens, and it is also very suited for creating portraits. In your update you mention you find the 350mm range important, and indeed, with the crop factor of the DX body, the 200mm will become 350mm. Another advantage you shouldn't forget is that the quality and amount of mega-pixels is high enough to do some serious cropping and still be able to print without visible loss of quality. The downside of this lens is that you have to switch your lenses for a wide shot, but the other option is to get the 18-200, but this is a much more expensive lens.

Both these lenses are the "cheapest" AF-S with a very good reputation on several review sites.

As already discussed in the comments, you're just beginning with DSLR photography, so in my opinion, try not to spend too much on a very high quality lens, but first try to find your personal style and preference. If you have this, you can base your next purchase not only on advice online, but also on your personal requirements.