Since the battery is new, it seems doubtful that it is not capable of holding charge. As Klaus noted, there may be a problem with the alternator and charging circuitry. I don't know where you are located, but many of the chain discount autoparts stores (e.g. AutoZone, Pep Boys, Kragen, etc.) have the ability to do some level of alternator checking right in their parking lot (not to mention battery checking). Of course, they are in the business of trying to sell batteries and alternators, so it helps to have done enough of your own "debugging" to be able to determine if they are giving you good advice.

I don't quite know what to make of the statement that the problem seemed to not be there for about a month after the battery got a good trickle charge, but that then the problem returned. Perhaps it is true that short-trip driving is not charging the battery long enough to keep it charged. But it really shouldn't take much. The other possibility is that something in the car is being left ON after the ignition key is pulled (not like an interior light, although that would do it, too, but something in the internal wiring).

This can be hard to figure out. There are multiple relays under the hood that are used to remove power from various circuits when the ignition key is in the OFF position. In theory you can find all of these relays, and check for voltage being present on the "output" side of the relay when the ignition is OFF (i.e., when the realy should be open, and therefore not providing voltage to the circuit). But this may require a detailed wiring chart, etc. You can do some degree of checking by just verifying that functions of the car that should be off are indeed off. But some are not that apparent. I had an intermittently sticking relay in one of my cars that powered the fuel injection control module. It was being left on every once in a while. It had no bearing on the car working, but the fuel injection controller circuit was on all of the time, and would drain the battery after a couple of days of sitting idle.

The one basic check that is easy to do is to verify that if you leave the battery itself disconnected, that it can hold charge for longer than a few days. But this may have some side effects, since some of the electronic control circuits within your car are intended to be left powered when the car is OFF, and some of these may require some level of resetting after the battery is re-connected.