I guess you can easily measure the expression of certain resistance genes by pulsing a group of cells or bacterias or fugi or whatever you want and comparing the pulsed group vs a control group at the transcription level in a timer course experiment.

You should design your own primers against resistance genes mRNAs, test them to ensure proper specificity and if they're quantitative enough through qRT-PCR and then extract total RNAs for each group at each time point for reverse transcription and qRT-PCR. You should select the proper normalizing gene for the normalization of your gene expression and then you can compare each time point vs control and also vs one other time point in the same group.

If you've related antibodies you can also perform western blotting time course analysis for the upregulation of these genes at a translational level.   

simple way is RNAseq, you compare whole transcriptome change.

If you don't have informations about what genes are responsive to your antibiotics yes, whole transcriptome microarray or RNAseq are a gold standard.

But if you already know what genes are involved, then spend founds in expensive RNAseq experiments doesn't worth the cost.

qRT-PCR is the good ways if you have specific genes and if you want to check globle change then go for whole genome transcriptomics.If you have seen some phenotypic change in the bacteria then target the gene related to that phenotype and compare with resistance bacteria if you have.Where is the problem plz specify you question.

By using RT-PCR with specific primers, it will be more accurate