Dense suspensions of model hard-sphere (HS)-like colloids, with different particle sizes, are examined experimentally near and in the glass state, under shear and extensional rheology. Under steady shear flow, we detect both continuous and discontinuous shear thickening (DST) above a critical shear rate (or shear stress), depending on the particle size and volume fraction. Start-up shear experiments show stress overshoots in the vicinity of the onset of DST indicative of microscopic structural changes, while the sample macroscopically shows dilatancy effects. Measurement of shear and normal stresses together with direct video imaging of the sample probes the appearance of the positive first normal stress difference, N1, at the onset of shear thickening at high shear rates and glassy states. This is followed by dilatancy effects accompanied by large fluctuations of shear and normal stress and stick-slip phenomena. Similarly, under extensional flow probed by capillary breakup and filament stretching setups, we find liquidlike response for low strain rates, while above a critical strain rate, the samples exhibit a solidlike behavior where thickening is accompanied by a macroscopic dilatancy and granulation. Monitoring the filament thinning processes under different conditions (volume fractions and strain rates), we have created a state diagram where all responses of a HS suspension (liquidlike, shear thinning, shear thickening, and dilatant) are shown. We, finally, compare the shear thickening response of these HS-like suspensions and glasses in shear with that in the extensional flow.