I travelled by myself last summer when I left home at the end of June to pop half way round the world and visit my best friend who had been studying a semester abroad in Auckland, NZ. I felt confident that I could navigate the airport and connecting flights, but i still wrote a comprehensive list of what time each flight was, what time it was boarding, which terminal it left from, and which gate (if i could find the information online!)
Although i was later than i would have liked buying my flights, I purchased them about three weeks before the fly date, they weren't too expensive. Make sure you're in constant contact with your American employer, and ask if the start/arrival date is flexible. There is a massive price difference between certain days, times, and flight durations, and thankfully my employer was flexible, so i was allowed to arrive two days later than i said i would when originally accepting the job. This saved me £200, which was useful for the first few weeks of living expenses.
There have been studies which prove that Wednesday is the cheapest day to fly, due to most holidays wanting to start/finish on the weekend, with Friday and Sunday being the most expensive. Flying early morning (yes, that means being at the airport at the crack of dawn) is usually cheaper, but it also means you can sleep once your on your flight. Take a look at slightly different duration flights also if you want to save. For me, it was cheaper to get a slightly later flight, but with a half 40 minutes less time in between my two connecting flights. Shop around, and use the STA travel price match, because if you find a cheap deal online with a non-budget airline, they will match it for you.
Make sure you have all your documents, i bought a handy document wallet for only £2.99, and it meant i could keep my passport, visa forms, money, and important bits of information all in one place inside my carry on luggage. In the week running up to your flight, make a list which you can keep adding to as you remember things you need to bring, because I can guarantee if you leave packing till last minute like I did, you'll remember that there are things you wanted to pack, but you wont remember what they are!
Things to make sure you've brought with you;