My 2002 Malibu battery drained

I just bought my wife this car, and I started noticing wierd problems.  First it would very subtly jut forward while accelerating(almost as if I was taking my foot off the gas and back on, however so subtle that I could see no change in rpms).  Next I notice that about half the time its seeming to have a hard time starting.  AND then my wife tells me today at work that the radio still works with the key out of the car!  She drives it home and says she can't get the key out of the ignition(in off position).  By the time I got home the battery was completely dead and the key just slipped out of the ignition.  I really don't know if the two problems may be related and I can barely afford to take it to a mechanic to have him chase the problem around.  Any help would be greatly appreciated... Really.  Thanks!