Sequencing using various Illumina sequencers is the predominant method for generating large amounts of sequence data. Generally, Illumina systems produce short reads (up to 150bp from each end of a DNA fragment). For many applications, particularly RNA-Seq or identification of SNPs or small indels in genomic sequences. If you are working on an organism that already has a high quality genome sequence, then it is likely that this will be the technique that you will use for your experiments. (If you work on something whose genome sequence has yet to be determined, you might want to look at nanopore sequencing)
The predominant use for Illumina sequencing by users within the Department at the moment is RNA-Seq, that is, determination of the expression of all of the genes expressed by the system under study by seqeuncing random fragments of purified cDNA from a number of samples. By counting the number of fragments that are sequenced from each gene, the relative amounts of a particular mRNA between samples can be estimated and patterns of co-expression can be determined.
Illumina machines are used to estimate species abundances in microbial communities, by PCR amplifying variable regions of their genomes (usually the 16S rRNA sequence).
We don't have any Illumina machines ourselve, but we do undertake all of the quality control and library preparation that are required prior to sequencing, before send samples to be sequenced either in Leeds (on a HiSeq 3000 machine) or to the Biorenewables Development Centre in Dunnington (on their MiSeq system).