I've already asked which wires I should use, and I believe I have that covered now, but I was also wondering what solder / specs on the solder are considered safe when attaching my wires to my heat bed. I have some solder here at the house, it's quite new (never been opened) it's Radio Shack
64-008 E 60/40 Standard Rosin-Core with 0.062 diameter...
That solder sounds like it should be fine.
Any solder should work, the only difference you might find is some of the older solder will include lead. if you were overly concerned about safety you might want to check your solder doesn't contain lead. However, for the amount you are doing, even a solder that contains lead isn't going to be an issue.