Whole-genome sequencing has facilitated genome-wide analyses of association, prediction and heritability in many organisms. However, such analyses in bacteria are still in their infancy, being limited by difficulties including genome plasticity and strong population structure. Here we propose a suite of methods including linear mixed models, elastic net and LD-score regression, adapted to bacterial traits using innovations such as frequency-based allele coding, both insertion/deletion and nucleotide testing and heritability partitioning. We compare and validate our methods against the current state-of-art using simulations, and analyse three phenotypes of the major human pathogen Streptococcus pneumoniae, including the first analyses of minimum inhibitory concentrations (MIC) for penicillin and ceftriaxone. We show that the MIC traits are highly heritable with high prediction accuracy, explained by many genetic associations under good population structure control. In ceftriaxone MIC, this is surprising because none of the isolates are resistant as per the inhibition zone criteria. We estimate that half of the heritability of penicillin MIC is explained by a known drug-resistance region, which also contributes a quarter of the ceftriaxone MIC heritability. For the within-host carriage duration phenotype, no associations were observed, but the moderate heritability and prediction accuracy indicate a moderately polygenic trait.