How can I repair/replace tires on a hand cart?
Quick and dirty
You could try some Fix-a-Flat. If nothing else, it might show you where the leak is.
Finding the leak
Pump the tires up and apply some soapy water to them, or hold them underwater in a large bucket.  This will help you locate the leak(s).
Plugging the hole

If the leak is in the tread, you could try a tire repair kit like this


If the leak is in the sidewall, you'll have to replace the tire.

If the leak is near the rim (and the rim is not bent), remounting the tire might fix the problem.