Short answer: yes, it's normal.
I know she's your new baby, but be firm and quick and no harm will occur.
Long answer: The neutral to first shift has the greatest difference in the speeds of the affected gears, (engine running-transmission stationary)hence the "lurch" you're experiencing.
You may reduce it a bit by letting the idle speed settle down before
that first shift, but don't try to "ease " it into gear as it can damage the gear engagement. And to answer the next question- no it won't get better with more mileage