Maybe you could think a little outside the box, and reconsider selling your kit lens. Bear in mind other than the 17-70 suggested all other suggestions are FX lenses, and other than the 18-35G suggested, you will have significantly narrowly FOV (field of view) at 24mm FX = 36mm DX, might drive you crazy when you need the extra FOV.
If the kit lens is failing badly in your book and you can't stand using it for another day...
- sell it
- you are likely to need another standard zoom
- the last I checked, most standard zoom that are not the top lenses (the f/2.8 Nikkors) are mostly "slightly sharper" than the kit lens, i.e. the loss of use of kit lens and the expense of buying a Sigma or Tamron may bring so little gain that it may not be worth the exercise. Plus Sigma's have well know focus consistency issues with Nikon bodies.
- That is, there will be quite a bit of loss (monetary) but not much gain (IQ).
- Then the only option to get significantly better IQ, IMHO, is to spend big and get the top lenses. Bear in mind to cover the same range you will need at least 2 lenses, say 18-35G and 24-120G (Nikkors I am suggesting) and these 2 lenses will set you back ~$2500. These are FX lenses that will be useful if you upgrade to FX, which may or may not be necessary.
There will be many times the kit lens is sharp and flexible enough to meet most purposes.
I second the other bros who suggest going prime route. Keep your kit lens for flexibility and non-critical applications, after all, it will bring back $180-$200, hardly worth the exercise.
Get either 50/1.8G ~$300 (a bit too close in FOV to 35) or 85/1.8G ~$600 and you will open up a new horizon in optical performance without spending megabucks.
Or explore ultrawide by going 12-24G or 10-24G, play with tele by going 70-300VR or 70-200VR (f4 for affordability), both options are ~$1000+ and you are opening new horizon. Ultrawide are DX lenses, telephotos mentioned here are FX lenses.