A helpful strategy is to cross cancel. When you cross cancel you are simplifying the fractions before you you multiply. This makes the multiplication a bit easier and less cumbersome. To cross cancel you look for a common factor that is share by any numerator and any denominator in the problem. You can use any numerator and denominator because of the commutative property of multiplication. Once you have "pulled out" or divided by as many common factors as you can, you can then multiply the numerators and the denominators that are left. If you have cross cancelled correctly, your product is usually simplified.
When you multiply mixed numbers you need to convert the mixed numbers to improper fractions first. You need to do this because you need to account for the whole value of the set. Remember 2 * 3 means three sets of two, so 2 1/3 * 3 1/4 means three and one-fourth sets of two and a third. You can not multiply the whole numbers separately from the fractions.