There are a lot of reasons your bounce rate could be high. Without seeing the site it’s tough to make suggestions on specifically what to improve first & many people have given good suggestions on what might be the cause.
One thing I’d add is that it’s hard to be sure you’re improving the situation without a controlled experiment. The best way to do this is probably with an A/B test. For those who don’t know, an A/B test is where you provide two variations on the page; they’re shown to users randomly and the winning design is the one which has the best result for some particular metric.
In this case the metric to improve would be bounce rate. I did some checking and it seems the content experiments in Google Analytics (free) allow you to have bounce rate as your metric of choice…
… I’d recommend you try some of the suggestions people have posted above, but measure the results and keep applying the lessons learned.