Yeah, sounds like you have anxiety - pretty much everyone with an anxiety disorders gets anxious about feeling anxious. It's basically an annoying spiral which keeps the anxiety going.
Sorry, I'm tired and horribly inarticulate, but I know how scary it can be but loads of people have it and it's not difficult to learn to control (though it can take some time). Cognitive Behavioral Therapy can help you recognise and correct irrational thought patterns. I would definitely recommend seeing a doctor - it'll reassure you more than anything.
Just remember you're not going insane and you're not going to have a heart attack (heart palpitations can be sketchy if you've never had them before but it is just anxiety).
A good technique to distract yourself when you're feeling anxious is to choose a 3 digit number and count backwards in sevens. Sounds silly, but it works.
Also, a technique my therapist taught me is to:
1. Write down the thought that's bothering you the most
2. What percentage out of 100 do you believe the thought to be true?
3. Write down all the evidence that suggests it is true and everything that suggests it is false
4. If your friend thought the same thing would you think they were being irrational?
5. How likely is it to be true?
Hope that helps!