What are the best ways of treating anxiety and panic attacks besides exercising or using any drugs?

Breath… deeply. think relaxing thoughts. skip caffeine. but excercise and medicine are the best.
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You could try blowing on a dandelion and making a wish. You have to work at these things. There is no easy answer. Take the drugs or do the therapy and exercise. That is what works.
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meditation, prayer, slow deep breaths
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Anxiety is self-created.

Nobody imposes anxiety. It is neither inherited.

It is you, who have loaded heavy anxiety on your small head. Unknowingly you must have added the weight slowly. Now the weight is more and unbearable. You should now know how to unload that weight of anxiety. Unloading is also to be done slowly and steadily. It cannot be done overnight.

Analyze how you have loaded this anxiety on your body. This anxiety comes due to unnecessary, unwanted and undeserving desires which we start developing. When the desire is not getting fulfilled, one gets frustration. Frustration leads to worry. Worry leads to anger. Anger leads to indecision or wrong decisions. One becomes restless. Mind becomes stressed. One does not get proper sleep. Then we try several methods to come out of it. We commit more mistakes by thinking that drugs and alcoholic drinks may reduce the tension. Sleeplessness leads to mental functioning getting affected. Mental upset, mental disorder. Our efficiency goes down. Our effectiveness goes down. Our memory weakens. More and more anxiety comes. We become a laughing stock before others. We take loans and we are not able to pay back in proper time. Family is worried and affected.

Can you now please see what is the origin of this anxiety. Cut the source now itself. Then only you can come out of this trouble.

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Negative emotions (like sadness, stress, anger, etc.) causes your Serotonin production to be low; when your Serotonin level is low, you are more prone to getting Anxiety, Panic Attacks, Depression, etc.

Medication like Antidepressants (SSRI – Selective Serotonin Reuptake Inhibitor) helps to boost Serotonin level.

But there are natural ways to do it without medication. There’s this strange herb called "St John’s Wort" – it is said to be more effective than Prozac. No, it is not for mild depression only and ignore those sayings. In fact, it does help anxiety and panic-attacks as St John’s Wort works like prozac. Other natural ways will be exercise, diet, more exposure to light, etc.
The problem is that, even if your Serotonin is balanced… you have that "learned behavior" in your mind. You need to break that initial cycle to destroy that learned behavior – Cognitive Behavior Therapy (CBT) does this. A technique that you can use without CBT will be Distraction… There are several other techniques to help cope them!
Ok, to use Distraction: Firstly, try to….

Extracted from Source.
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http://PanicAttackResearch.blogspot.com

Try Bachs Rescue Remedy…Its Herbal And Really Helps Me
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Stopping panic attacks can be alleviate by any of the following simple techniques.
1. Breathing deeply through the nose and out through the mouth ten times before you stop.
2. Counting down from 100 to 0, tricks your mind into thinking you are lowering the panic level as your count down.
3. Visualize yourself somewhere, say on holiday where you felt very relaxed and at ease.
4. Change your thought patterns. If you think you are going to panic you will. Conversely if you see yourself ahead of time, going into the situation with confidence and being in control this will greatly improve the experience you have.
References :
http://backtohealthguide.com/category/stress-and-anxiety/panic-attacks-stress-and-anxiety/