Last updated January, 2017.
"My goal for meditating is to overcome my anxiety. Is this reasonable?"
First, consider changing your perspective - instead of "overcoming anxiety," make it to deal with anxiety skillfully.
This is important because if you believe you'll completely overcome something (i.e., get rid of your anxiety), you'll be frustrated if anxious thoughts and emotions come back. You will most likely judge yourself for not being successful (for failing), and will probably lack the motivation to continue your meditation practice.
Second, what does the word "goal" mean to you? Are you rejecting your present circumstances, driving towards a future vision of yourself, and pinning your happiness to whether or not you achieve that vision?
If you're wishing your circumstances are different than they are, this will likely give rise to more anxiety and make your condition worse. Instead, accept things as they are right here and now. No amount of lamentation will change your current reality - only skillful action will.
Begin by cultivating awareness of what's happening in your mind. See what your anxiety is composed of - is there doubt? Fear? Worry? What are the physical sensations - rapid breathing? A racing heart? Tension in your neck and back?
Notice what's there, but don't try to fight it - simply let it be as it is. And then, move your attention away from it. This may be uncomfortable at first - we tend to feel we are "giving in" if we aren't fighting. But, that's not the case.
Those thoughts and emotions are there whether you fight them or not. And, if you do fight them, you are indulging them: you are giving them fuel. This gives rise to more thoughts and emotions, and they become persistent and pervasive. Before you know it, you're being dragged into a "pit of despair."
When you let everything be as it is, however, you will discover it's not the problem your mind made it out to be. The thoughts and emotions are part of an unfolding process, not an isolated event that looms before you, threatening your well-being. The thoughts and emotions are contingent and temporary: they come (arise) and they go (pass), provided you don't allow your attention to get caught up in them. Provided you don't indulge them and give them fuel.
How do you do it?
When you meditate, focus your attention on an anchor (usually your breath or a mantra). When those thoughts and emotions arise and you notice your attention is caught up in them, simply guide it back to your anchor.
Do it over and over, because you are developing a skill. And, you can transfer this skill to your "non-meditating" time.
When you aren't meditating (when you are going about your daily activities), focus your attention on the present moment. When those thoughts and emotions arise and you notice your attention is caught up in them, simply guide it back to the present moment.
This is how meditation and mindfulness work.
You aren't "overcoming" anything, you're changing your relationship with it. You are training your attention to no longer get caught up in - and dragged around by - a compulsive mind and its non-stop activity.