How to measure quality and quantity of DNA?
You can run your DNA sample on agarose gel to see, whether you have significant degradation.

If you are interested in contamination, you can make a standard photometric analysis to assess the 260/280 and 260/230 ratios and absorbance at 320 nm on NanoDrop or even something similar to Eppendorf's BioPhotometer.

In case RNA may be an obstacle for some down-stream procedures, with Qubit you can also measure separately DNA and RNA concentrations and then decide if you have too much RNA and it's worth treating your sample with RNase.