this is in michigan, but it should be similar.
you can't file for unemp until you have actually been laid off, so go in the first day that you can, to be processed, and try to apply by friday of the week, they will count the applying week as your starting week.
michigan had a phone system, called marvin which was used to make your 2 week claims. most states do it by mail, and have actually met someone that had his sister mail in the claims for him, while he was out of the country. (illegaly btw ^^) i dun think you can leave the state, and you have to be "actively" looking for a job which is about 3 apps, can't turn down any jobs, so apply for positions that you likely won't get, and hey if you do, cool right? can't goto school, must be available to work anytime. (i told them i was going to school at the time, and this delayed my processing 🙁
you can get a part time job, but 50% of your pay will be deducted from your benefits, so think about that before picking up a job. they say you can not claim a week if you were paid a bunch that week and will lessen your benefits pay. (the unemployment benefits are measured by weeks paid, rather than amount paid out to you, so one week of benefits pay, either 100% or only 10% of what you are elligible for, you now have one less week of benefits. (not sure how to skip tho.. never had that problem ^^)
sorry to hear about what happened, but hey, free money is always good right? btw, i think you get 75% or something of your pay at the time of termination upto a capped amount. and you can file for an extension after the first 6 months, or 26 weeks of benefits. (at least in michigan..) and i think you have to work for at least a year before you are elligible for benefits, but this may vary by state.
if you got other questions, feel free to pm me.