How many times per day should my baby eat, and how should I deal with any feeding problems?

Weight gain is key. Feeding patterns of infants vary with age, and from one baby to another. The bottom line is that growth/weight gain should be adequate, and the baby should appear well nourished, happy, and active. As far as any feeding problems, that can mean many different things, and i'd be happy to hear about specific questions or challenges you are encountering.
Free demand. Newborns may eat 10 to 12 times per day if breast fed. They grow very rapidly and require a great amount of calories. Feed the baby on demand and you can not go wrong!
When they are hungry. As infants you should feed your baby whenever they are hungry. Generally this is every 2-3 hours. As they get bigger they will eat more with each feeding and the time between will lengthen. But even 3 and 4 mo babies will eat every 3 hours or so during the day. Count number of wet diapers if there are not 6-8 daily see your dr. If you are concerned about any feeding problems see your dr.
About 6 times a day. Put simply, a baby's eating times are: breakfast, lunch, dinner, snacks between breakfast/lunch and lunch/dinner, plus a late evening feeding after dinner. Babies under 3 months usually wake up to eat one additional time, between 11:00pm and 6:00am. Parents should remain flexible on feeding times, as babies don't live by the clock.