Start date for work in Berlin?

No. You will have to pay German tax on all income world-wide as of the date you start your German residency. The date and not the location of where the money is earned are decisive. You will then have to file both German and American tax forms annually. The US is one of only 2 countries that have this requirement. Whether you will have any tax liability to the States once living abroad depends on a number of factors. You'll need to look at the possibility of Foreign Paid Tax credits or taking the Foreign Earned Income Exclusion. I have posted in detail about these subjects on other threads, otherwise Google it or look at the irs.gov website.

Tom is right above, but I would like to add the following:
The income tax rate applicable to you will be calculated based on your (worldwide) income of the whole year - the more you earn, the higher the percentage is. The calculated rate is then applied only to your (worldwide) income for the time you reside in Germany.
Thus you have to declare all year's income in your German tax declaration, and non-taxable income in the time before you came will still increase the tax you pay on your income after moving. This is called "Progessionsvorbehalt".
The whole thing is very complicated and you should engage a good tax consultant (with experience in international tax matters!) for your declaration - the fess he/she takes are well spent!

What Beppi wrote just reinforces what I did; the timing and not place determines which income is reported where. If one has a job and gets a normal paycheck then the tax filing is not really so complicated in Germany. There are clubs to help people with this for free or maybe a small membership fee. Look for “Lohnsteuerhilfe”.  If one has their own business or is self-employed then it is more complicated and there is really no free help available that I know of.  What makes taxes complicated is the language. If one doesn't have good German then they probably have little chance – and like elsewhere there is specialized vocabulary that even the average native speaker might not know.

But if one does have good German, I suggest having a professional do the forms the first year and then one can fairly simply copy what they did with any number of tax preparation programs. Should one bother? For me yes. I had someone do my taxes for the first 2 years and they charged me around 10% of my annual income just to fill out forms and do a basic book keeping that I was doing anyway for my American tax filing. One sees advertisements for tax preparation and how much money one will save. This is assuming that people are total idiots and don't take even the most basic deductions. That one's tax preparer is going to find lots of hardly known deductions to save one lots of money is an illusion.

I have a background in Finance so I have a better feel for such subjects than most but I have an Argentinian friend for example who is self-employed and also does his German taxes himself. His German is good and he has a good head for business although no formal education in such a field. And don't let the Finanzamt or anyone else con you into thinking you are required to hire a professional – simply not true. I honestly don't know how people afford to hire someone to do it every year since it is so expensive for self-employed people. If one is on a payroll, then the filing is not so complicated but on the other hand not so much work for a professional to do - so maybe affordable.