The line that goes from the shutoff valve on the wall to the bottom on my toilet tank has a leak in it. It is an old pipe that I believe is copper on the inside and silver on the outside. Can I replace it with a newer bendable line, or do I need to be worried about it leaking when I'm not home? Is there a part I should be buying to go along with the new line?

Yes. You will need the plastic ferrules and compression nut or you can use the flex lines.

Recommend you purchase a braided flexible supply line. There are two types. There are faucet supply lines and toilet supply lines. Make sure you get the toilet supply. Flexible supply lines are much easier to work with than the hard lines, which use compression fittings. The hard lines also require that you shape and cut to proper length. No reason to be concerned about leaking or bursting.

You can use braided flexible supply line as harold says, or you can use a plastic supply line see the image below, Thanks.

John