First of all, as many major agencies point out with regards to Passover, if it's a heart medication, antidepressant, or any other medication without which there would be serious risk to one's life; take it first and ask kosher questions later. If you can get one with better ingredients kosher-wise that's great, but first and foremost make sure you're taking it.
There are different opinions and practices with regards to pills that are swallowed; if I recall correctly a major opinion is that if someone is "seriously but not life-threateningly sick", then the rabbinic prohibition against swallowing pills with non-kosher ingredients would be waived. Hence someone with bad arthritis could, for instance, take chondroitin pills even if the only available ones were made from non-kosher fish.
For someone who is just "feeling a bit something" (e.g. a small headache), there may be less room.
Similarly, there are dispensations for medicines that may not apply to vitamins that a healthy person takes for optimum health. Hence many a great rabbi requires a kosher certification on multivitamins (of which there are now several brands, fortunately, and many are easy to find at vitamin or health-food type stores).
There is another rule that with small children, increment their level of sickness by one. So a toddler with "just a headache" should be given the dispensations of a healthy adult who is now seriously knocked out; and a toddler who is "seriously sick" could allow for things otherwise reserved for "serious risk to life."
Chewables and syrups are more of an issue, as you're ingesting them like food. A common concern is glycerin in many syrups; glycerin could be from animal or plant sources. If you can't find a kosher-certified version, the Star-K has recommended diluting the syrup with several parts water, so that any non-kosher glycerin is less than 1/61 by volume of the total mix. (While we frown upon taking something you know is non-kosher and diluting it, here it's reasonably likely this glycerin was plant-derived.)