Is it safe to give my baby cold or cough medicine?

No. Most children under 5 years of age (definitely under 2) should not take cold medications.Exceptions may be made for some cough medicines under the guidance of your physician. In general, this is because there may be serious side effects to those medicines in young babies (and oftentimes they do not work). Try saline and a humidifier. Honey if your child is over 1 may also be beneficial.
No. Cough and cold medicines aren't for babies-- they don't work, and even worse: they're not safe.
No. Plain Acetaminophen (tylenol) can be used for pain or fever (call your doctor first if your baby is less than 3 months). If your child is older than 6 months you can also use Ibuprofen (advil/motrin). But over the counter cough and cold medications are dangerous in children less than 6 years old.
No. Years ago, it was common for 3-12 month old babies to take cold or cough medicines. These medicines were smaller doses of adult cold medicines. Research has found that these medicines often did not relieve babies' symptoms, and that some babies were getting excessive doses, sometimes harming the babies. Nowadays, the fda and physicians do not want babies to use cold or cough medicines.
No. Not really. Cough and cold medicine has many side effects, is not well tested in babies, and is difficult to dose. It does not treat the cold or help it get better any faster. Because of the risks, most pediatricians do not recommend these for young children.