Car won't start (but has new battery) What could be wrong?

I have a front-wheel drive, maunal transmission 1992 Hyundai Excel, and it will not start (all the time).

I recently (approx. Two weeks ago) had to replace the battery, due to the old one being flat. All of the connections to the new battery are tight and clean.

When I place the key into the ignition and turn it, no lights on the dash board come on (ie the oil light, the battery light, and the other light that usually comes on). However, if a door is ajar, the light comes on, and is quite bright.

The car makes no sound at all when turning the ignition key. (Unlike the churning, whining noise it makes when the battery is flat and really trying.)

Two days ago the car would not start. This was within 1/2 - 1 hour of it being in use.

Today, I had to have the car jump started. But, yesterday and this morning the car started fine.

When I got the car home, I was checking the lights and the sound of the motor, and when I put the headlights on, the car died (the engine was on and running at the time).

Recently, the indicators have been ticking very fast (I don't know if that is relevant).

I do not usually have the radio on (as it has been removed for over a month), do not have power windows, do not use the heater or fan, and do not have air conditioning. Although, I do have power steering.

I have checked the belts, and they seem fine.

If you need to know anything else in order to help, ask I will tell you in detail as much as I can.

Thank you for any help you can give me :)