Should I solder wires to the connection points?
Looks to me that your breadboard probably has a break in the +5v and ground lines, right where there is a break in the blue line next to the ground line. However, this may have saved you from burning out the backlight, since most LCD displays need a resistor in series with the backlight. If you don't know the backlight power requirement for that particular LCD, try a 150 ohm series resistor.
I suspect that the resistor you are using is much higher than 150 ohms, or that you have connected it incorrectly. Can you post a photo? Also, if you can post a link to the lcd datasheet, I can determine whether a resistor is needed and what value it should be.
That's a 1,500 ohm resistor. You need Brown, Green, Brown, not Brown, Green, Red.