There are 2 issues that cause problems when experiencing anxiety.
Firstly the physical feelings of anxiety, the dizziness, churning stomach, sweating etc. and secondly there is the psychological stuff, the thought processes, the worrying "What if I panic?" etc.
Both of these need tackling in order for a good resolution to your issue.

The physical problems are because of what is called 'The Fight / Flight Response'.
This is an instinct inside of us all that has been with our species for millions of years, it's a way of boosting muscle strength in order to either run away from or fight our attackers. Unfortunately in the modern age we don't need this instinct very much as we are no longer being chased by sabre toothed tigers but our instincts don't recognise the difference between a real threat and an imagined one. Or even the difference between something that's exciting and something that's scary.
Obviously there are many levels of apprehension, I use the analogy of a ladder.
We need to bear in mind that any emotion at all that doesn't create a feeling of safety will instead create the fight or flight response to some degree.
Even being late for a bus creates the same physiological response as running for your life, just less of it.
When suffering with Anxiety we can jump from the very bottom of our ladder to the top rung in a matter of seconds and for what seems like no reason. Once your instincts get into the habit of experiencing the fight or flight response it can kick in with only the slightest hint of negative emotion.

To tackle this, you need to teach your instincts that everything's fine. That actually there isn't a mad man with an axe, or a sabre toothed tiger. Best way to do this is to learn how to breathe correctly.
The reason your body goes crazy when your anxious is because of the sudden need for fuel in the muscles, for oxygen mainly, and so tries to find as much blood as it can. There is always spare blood in the gut area and often that's the first place your body goes to find it and so you're likely to experience a pulling sensation in your tummy as the blood is moved out, this is often called 'Butterflies' and you've probably experienced this a million times in both nervous and excited situations.

So if you can get as much oxygen into your blood as possible then you won't feel so much of that pulling sensation, also your heart won't need to beat so many times to get the oxygen around your body and will slow down. This then gives the impression that whatever it was that was threatening you has gone and the danger is over.

Doing this is fairly easy, by ensuring that as you breathe in you push your belly out and fill your lungs right to the bottom you'll get a massively increased amount of oxygen into your blood and by holding it in for a second before breathing out slowly you'll slow your heart rate down.

Do this at least 3 times and you should find the symptoms decreasing in no time at all.
This is called diaphragmatic breathing and is something that I teach almost every client that I see.
Something else that crops up in most sessions is a technique vital for overcoming anxiety. It is called 'Thought Stopping'
You'll find a video about it HERE but i'll go into it briefly here too.
It's pretty much impossible to go from thinking about something bad happening to suddenly start thinking about something nice instead.
You need to put something in the middle of the process to interrupt the bad thought, as it will be very vivid.
Best way of doing this is to imagine the word STOP. Not just as an image in your mind but also imagine what it sounds like and as loudly as you can too. STOP!

Imagining the word stop in your mind as loudly and vividly as possible should interrupt the negative thought enough for you to move onto something else. Either the exact opposite of the original thought if needs be. Or something normal and trivial such as "Do I need any milk?"
There's a video explaining this on my videos page.

Practise meditation or hypnosis, there are some free downloads on this site to get you going with that. Subscribe to the Newsletter And you'll receive an email with a link to some.
Combining the mental exercises of thought stopping and hypnosis with the physical exercise of diaphragmatic breathing should set you well on your way to overcoming anxiety.