Just answering your question straight out: It probably costs around 100/hr (give or take depending on your location). As far as one that deals with personal issues, most "family therapists" can deal with most personal issues. It depends what your actual issue is. For example, some therapists specialize in areas like post traumatic stress disorder, rape, divorce, adolescent behavioral problems, etc.
If you are just having a generic feeling of dissatisfaction in your life and need someone to help you figure it all out, probably just a psychologist or family therapist can help you. My suggestion is that you just call a few you find in the yellow pages and ask them what they specialize in.
Back to the cost issue, many insurance companies have a "mental health" part of their plan. If you have insurance either for yourself or through your parents, you may be able to go for a greatly discounted rate or even just pay a co-pay.
One last thing is your question hinted at just one session... that is rarely the case. Most problems that cause us to seek help are deeply rooted enough that one session is not enough. It can take several months (or more) to figure out why you are actually feeling the way you are and then try to work on techniques to get you beyond that difficulty in your life.
If this is something you are able to financially handle I recommend it. Sometimes it just takes that one little question or piece of advice to change your entire outlook on your issues.
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