The Linksys power cable is too big for the Modem

Are you trying to power the modem with the router's power cord? You should NEVER use any other power supply than the one that came with the device. They are all different, with different voltage outputs, and current capability. Some have the positive and negative leads reversed and using a different supply can cause substantial damage to the device, and/or the supply.

Then again, when I plug in the Modem Power Cable, it doesn't give me the 6 lights.

You should still seem some lights.

You should be able power down everything (unplug power cords - including the computer's). Disconnect your router from the modem. Then plug in your modem and wait for the lights to stabilize, then plug your computer directly into the modem and power up the computer. You should get Internet access. If all is well. Power down and unplug everything again. Then connect the router to the modem, using the Router's WAN port, and your computer to the Router via Ethernet, and power up the router, wait for lights to stabilize, then connect and power up the computer, and see what happens.