Depends on the drink, really. Some drinks can fuck your stomach lining over more than others do, for example.
In a general sense though, the headaches and similar symptoms are often caused by simple dehydration. Lots of electrolytes (salt, Gatorade, food in general) and lots of water are the key things to helping out with that. Pure water isn't that great though, considering you do need electrolytes to be well hydrated.
Keeping food down is pretty important, but from my experience avoiding things like dairy helps, considering the last major hangover I had I vomited about 5 minutes after drinking milk. Could barely finish a small McDonalds meal but after I did eat it I felt a bit better.
Edit: I find the general key is to avoid being hungover altogether though. Pace your drinking, eat lots of food and drink lots of water WHILE you are drinking and BEFORE you sleep, and you'll wake up in the morning feeling quite all right I've found.