(2-part post):
You can claim JSA online here:
http://www.dwp.gov.uk/eservice/
Alternatively, you can claim by phone: 0800 055 6688 (8.00 am - 6.00 pm Monday to Friday).
Once you've made your claim, you'll be given an interview appointment at your local Jobcentre Plus. (Actually, it might not be that local. Only the larger Jobcentre Plus offices carry out interviews. You might have to travel slightly 'out of area' to attend the interview. If you do, make sure that you claim your travelling expenses, which will be refunded in cash the end of the interview).
There are two types of JSA. Since it seems likely that you'll have paid sufficient National Insurance contributions, you'll initially receive Contribution-Based JSA. The good point about C-B JSA is that it's not 'means tested' against your savings (or your partner's income). The bad point is that you don't get things like free prescriptions, free eye tests and free optical vouchers.