If your aim is detecting the presence of topoisomerases mutations associated to quinolone resistance, the best method is conventional PCR of the genes involved (gyrA, parc, gyrB, parE) and sequencing. As a first approach, you do not need to sequence the whole gene, mainly in the case of gyrA and parC, since almost always quinolone-resistance associated genes are located in a small region (quinolone resistance determining región, QRDR).

PCR, I thiink...

I am not clear about your research question: 'sequence types'. 

Are you looking for the resistance mutations and is the genetic determinant of ciprofloxacin sufficiently well known (as Juan indicatesin his answer) that you need indeed only amplify the one gene and have it sequenced? In that case classical PCR with standardised primers is indeed the most cost effective, especially if you can limit it to a region and need only a single sequencing reaction per sample. But if the research question is to genotype the E. coli outbreak to determine strain, origin, diversity etc this is a very different thing and would require the analysis of multiple genes both associated and not associated with the primary target of the drug. I that case you may well find that next generation sequencing of the whole genomes is a cost-effective and fast option nowadays. You can shop around and ask for a quote from what is now a rather large number of suppliers of such services. 

Thanks, your answers have been quite insightful.

I should have specified that we are interested in determining what proportion of our isolates belong to the ST131 clonal group and then identify the ones that don't. We are currently thinking of using PCR to amplify the pabB region as a first step approach to identify the ST131 clones.