Well done for getting yourself a new bike, beware cycling is addictive.
Gears.
The bigger the front 'chainring' and/or the smaller the rear ring on the 'cassette' the higher the gear, which means the further you travel for one turn of the 'cranks'. Great for high speed if you are strong enough to keep it turning over fast enough, but very hard work up hill.
Conversely the smaller the front ring and/or bigger the rear the lower the gear and the distance covered by one turn of the cranks is smaller, this helps you get up hills.
You say it's a road bike, so it wil have a choice of two rings at the front and 9 or more at the back. As the chain has to flex sideways to run across these rings it is advisable not to run the big ring at the front to the biggest 3 or so rings at the back, nor the smaller front ring to the smallest 3 or so at the rear. There is a huge overlap between the ratios available from all of the combinations so this will be no hardship. Doing this will reduce chain wear.
Freewheeling is when you stop pedalling and the bike continues on its way with no effort. The bike slows down but not by much if you are on a level road and the bike will continue to accelerate down hill.
Grinding and Spinning refer to 'cadence' which is measured as the number of turns of the cranks in one minute.
If you pedal too slowly you are grinding and your knees will not thank you for it in the long run!
If you pedal quickly you are spinning and this is regarded as a good thing. It is more efficient and kinder to your knees.
Cadences of 70- 80 are typical of most experienced cyclists, racing cyclists spin faster, nearer 100, there are exceptions! Big hills will eventually reduce everyone to grinding depending on gear ratios fitted to the bike and rider fitness.
Hope this helps, don't be afraid to come back with any more questions. Also Google Sheldon Brown. The late Sheldon left a huge archive of material about cycling for use by novices and experts alike.
Cheers byegad, aka Geoff