Train to grindlewald from Zurich

Yes. If you do maps.google.com and put in Zurich Airport, you'll see the train symbol right near the terminal (near a Migros market). Plus, it says on Google: "Airport with domestic & international flights, 3 terminals, a railway station & bus links."

Find the ticket office for the trains near there. They'll speak English. Say you want to get to Grindelwald, BUT ask to be routed via Luzern as it is a more scenic route. You can even stop awhile in Luzern (Lucerne) to look around, as you have to change trains anyway. If you ask, the agent will print out a slip with the times and tracks listed on it. You might even say that you might stop for an hour or so in Luzern, and the agent can give you times for the following trains (generally every hour).

There are storage lockers on the lower floor if you don't want to lug your luggage thru Luzern. Still, even if you have a few minutes between trains, it's worth getting up to street level to take a look as you are right near the lake. You'll see a view pretty close to this:
http://tour.luzern.com/360/en/

You can plan all sorts of train excursions from Grindelwald, though I'd wait until you get there to get tickets (rarely any discount for advance purchase). Besides, you might just decide you want to explore the Jungfrau area around Grindelwald. There are plenty of sights and trails. If you click on my name you'll find a link to a website where I describe our dozen favorite trails in the area, with maps and pics.

km,

Yes, the train station at Zürich airport is right within the airport. As I recall, there are two ticket offices, one for "long distance" trains (ie: to Italy or wherever) and one for trips within Switzerland. The ticket agents speak English, so you shouldn't have any difficulties.

If you're planning on taking lots of day trips, Grindelwald is perhaps not the best "home base" location. Lauterbrunnen is a more central location. You can check all your rail travel solutions using the Bahn.de (German Rail) website or the Swiss Rail website.

Your date format is a bit confusing but it looks like you'll be there for about two weeks? Having one of the Swiss passes might save you some money and you may find this website helpful - https://www.ricksteves.com/travel-tips/transportation/trains/switzerland-rail-passes . There are a variety of Swiss passes and choosing the right one is often a bit confusing. Unfortunately you'll have to do the math and figure out which one best suits your situation. The Berner Oberland Regional Pass is one you could consider.

How will we get to Grindelwald.....

• When you've collected your luggage, make your way to the airport rail station and buy your tickets to Grindelwald.
• The trip from the Zürich airport to Grindelwald will be ~2H:52M, with two changes in Bern and Interlaken Ost (Ost is the second station in Interlaken).
• When you arrive at Interlaken Ost, you'll change to the Berner Oberlandbahn. It looks like this - https://goo.gl/images/J4Ufdh .
  
• Before boarding the Berner Oberlandbahn, check for a small digital display in the window which will indicate the end destination. Some of the trains split in Zweilütschinen, with part going to Lauterbrunnan and part to Grindelwald.
• The trip from Zweilütschinen to Grindelwald is ~21 minutes. This is the Grindelwald station - https://www.jungfrau.ch/en-gb/arrival-at-station-car-parks/grindelwald-railway-station/ .

Have you travelled by train in Europe before and dealt with changes, etc.? I would highly recommend packing along the RS Switzerland guidebook, as there's a lot of good information there that will help your trip go smoothly.

Is the Zurich train station within the airport?

If you interpret that question literally, no.
Zürich HB is the main station in the centre of the city.
Zürich Flughafen is the station at the airport, directly accessible undercover from the terminals, "Flughafen" is German for airport.
Two different places. there are also lots of other stations in Zürich called "Zürich-something".

You can look up train times on the SBB website (SBB = Swiss Federal Railways): https://www.sbb.ch/en
You can enter From: Zürich Flughafen, to: Grindelwald, and a date and time, and find the trains. You will have to change trains to get to Grindelwald, but that is how the system works, it is an interconnecting network.

This is a map of the Swiss rail network: https://www.sbb.ch/content/dam/sbb/de/infotexte/uebersichtskarte-sts.pdf
All the red lines are rail lines, and every one has at least one train per hour. The map does not show all stations, there are too many of them, just the lines and main stations.