Lens suggestion: wide angle for vacation

There is no such thing as a good, cheap, wide-angle prime for DX. 24mm is not wide on DX, and 20mm and 16mm primes are bloody expensive.

If your budget is limited to €300, you can forget 2.8 DX zooms too (unless you find a Tamron 17-50 in mint condition for that money, in which case, you should grab it).

The best for you might be to get a 18-105VR. At 18mm it is f3.5, and combined with the VR, it should deliver you quite acceptable images. I know, it is not 2.8, but it is much wider than 24mm and that is the best bang for money for DX. The 16-85 is out of your budget range.