Can I continue to take antibiotics pills on a fast day (to be taken every 8 hours)? If yes, what is the best way it should be taken?

These pills may be taken if they are swallowed and not chewed.

Although it is preferable not to drink water while taking a pill, if it is needed to take the medication, one can drink a small amount of water to help the pill go down. If water is needed, the water should be made bitter first.

This applies to fast days other than Yom Kippur. For Yom Kippur, see here.