You would apply at the nearest U.S. Embassy. 6 months is the absolute maximum you can stay in the U.S. without applying for permanent immigration status, and while you re here you probably would not legally be allow to accept any type of work.
You need to fill the DS-156 form online and print it out afterwards; hand-written forms are not accepted. The DS-157 needs to be printed out and filled using a black ink ballpoint-pen. Since based on your age administrative processing is required which may take up to 60 days I recommend to apply for the visa early in advance. Visa issued in the United Kingdom are valid for 10 years, with multiple entries and a stay of 6 months can be granted for each visit. The fee is $131.I recommend to read the website of the Swedish Embassy; it has excellent information what type of documents need to be forwarded. http://stockholm.usembassy.gov/consulate�� ..For the appointment for a visa interview see the website of the London Embassy: http://london.usembassy.gov/cons_new/vis�� The waiting time for a visa interview is 4 working days at present.You may stay visa exempted in Canada and Mexico for up to 180 days.http://www.cic.gc.ca/english/visit/visas�� ..http://www.inm.gob.mx/EN/index.php?page/�� ..New Zealand also grants a six months visa waived stay for British passport holders.http://glossary.immigration.govt.nz/Visa�� ..For Australia you DO NOT need to get an ETA through a travel agent and pay the service fee of AU$20 [��10] and whatever the travel agent charges for the service.http://www.immi.gov.au/visitors/tourist/�� ::European passport holders can apply online for a free of charge e-visitor. The e-visitor is a label-free visa; you will receive an e-mail from the Australian Immigration Service stating that you are eligible to travel. The e-visitor is valid for one year and can be used for several visits and up to 3 months stay.http://www.immi.gov.au/e_visa/evisitor.h�� ..http://www.immi.gov.au/visitors/tourist/�� ..
You simply need to follow the directions for those forms on line, then you will get a page instructing you on how to make an interview appointment. At that time, you will get a page telling you how long the wait is for interviews and how long it takes after the interview to get your visa if you are accepted. The cost is exactly $131.00usd and you will be instructed how to pay that. Do Not fill out the forms until you are ready to go in for an appointment...appointment wait times are usually less than one month. You do not need a visa for Mexico, just a tourist card you will get on the plane or at the border. You do not need a visa for Canada for a visit of up to 6 months. You do not need a visa for New Zealand. See 1st site below and scroll down to the list that shows all the countries you may enter without a visa...they are on that list. You do need an ETA visa for Australia. See 2nd site. The fee is 10 pounds.