Common Metrics of Assessing DNA Sequence Quality
As mention in the comments Phred scores are the quality scores for most sequencing platforms. This value expresses the probability of a base is being called wrongly. You can find more information here. This values can be found in a fastq file coded by symbols.

For example, a phred score of 30 means that there is a probability of 1 base being wrong every 1,000 (that is 99.9% accuracy). In general values around 30 are considered good enough, but it depends on your analysis whether you want to be more more or less restrictive.

Biopython has some examples on how to plot and filter phred scores. There exits lots of other software to do QA checks, for example this one. 

You can also check length and distribution of the reads, duplication levels, etc.

In case you need to trim sequences from Illumina there are several stand-alone software like this that you can use.