My exercise instructor says we should learn to relax by emptying our minds of all thoughts, and then meditating on peace and calmness (or at least I think that's what she's saying). Does the Bible say anything about meditation?
Yes, the Bible does urge us to meditate, although not in the way your instructor is apparently recommending. It’s not wrong, of course, to want to be free of fears and anxious thoughts, but simply emptying our minds is not the solution; it may even open us to greater problems and worries.
Instead of emptying our minds, therefore, the Bible urges us instead to fill our minds — with God’s truth. Let’s face it: In the course of a day, all sorts of thoughts lodge themselves in our minds — some good, some not so good. Those thoughts can, in turn, give birth to all sorts of worries and anxieties; they may even lead us to do things that are wrong or destructive.
How will we conquer these thoughts? The Bible gives us the solution: Turn them over to God, and meditate on Him and His truth. Truth drives out that which is false, and there’s no greater or more reliable truth than the truth of God’s Word, the Bible! This is why I urge people to read a portion of the Bible every day — and not only read it, but also meditate on what it means and how it applies to their life. The Bible says, “Blessed is the one… whose delight is in the law of the Lord, and who meditates on his law day and night” (Psalm 1:1-2).
Set aside time each day to be alone with God — praying, reading His Word, and meditating on its truth. When you do, anxious thoughts will flee, and peace and trust will take their place.