Does free trade lead to lower prices?

Maybe.

I'm no economist, so I found this analysis of free trade to be enlightening.

When a country opens up to trade (or liberalizes its trade), it is the relative price of imports that comes down; by necessity, the relative prices of its exports must go up! Consumers are better off to the extent that their consumption basket is weighted towards importables...

...And in the U.S., the Wal-Mart effect has to be qualified to take into account the fact that the relative price of the goods that the U.S. exports (including for example agricultural commodities) is higher than it would have been absent trade.

So cheap things at Wal-Mart make food more expensive. Unless you just use the credit card, that is:

Of course, if you are running a huge trade deficit like the U.S., you can have cheaper prices all around—for all to go on a consumption binge as long as the party lasts.

The last thing I found interesting was that globalization advocates tout the economies of scale in large-scale production, but Rodrik puts the kibosh on that:

Scale economies raise a whole set of new conundrums (which is why I had stuck with the standard comparative advantage story). In particular, since scale economies are not compatible with perfect competition, we find ourselves in a second-best world with all kinds of strange possibilities.

In other words, free trade can give us cheap imports (Chinese lead toys), but may make our exports more expensive, especially if we stop having trade deficits.