If it started okay right after you put the new battery in and now doesn't
start after a couple of months, you might have a bad alternator that's not
charging the battery properly. Since the starter is a higher amp draw than
anything else in the car, it's very easy for a battery to be able to light
lights, sound the horn, etc. but not be able to move the starter. (That
"click" is the sound of the solenoid kicking over, which doesn't take much
juice. However, once the solenoid has activated the heavy gauge circuit to
the starter, there's not enough to operate it.) Even if you have not seen
any alternator warning light, it's possible for an alternator to put out
enough to keep the warning off even though the battery is slowly draining.
Also, the belt driving the alternator could slip silently, thereby not
spinning the alternator enough to recharge the battery.
First diagnostic step: Does the car start with a jump from another vehicle?