I bought a new drier with the "inverted V" 3-wire power plug. The drier outlet I have is like an overgrown version of a 110 grounded outlet, with 2 vertical 110 prongs and a ground socket. I bought a V-socket to accommodate the new drier plug, but when I shut off the breaker and opened up the old socket, I found red and black hot wires, and apparently a bare AND a white ground wire. My question: Do I need to hook BOTH the white and bare wires to the ground terminal, or just ignore the white wire altogether?

Well first off that plug and receptacle you had was wrong for a dryer, unless it was a true 240v like a European model. The receptacle you had was a straight 240v-30A version.

You should replace the receptacle with a 4-wire dryer receptacle and get the matching cord for your dryer.
The dryer does not come with a cord from the factory, you put the proper cord on it. So whoever put that three prong cord on it assumed it was correct.

Also, when you connect the dryer to a 4-wire receptacle, remember to remove the bonding jumper from the ground to the neutral.