What is the best way to travel from Grachen/Zermatt Switzerland to the Dolomites in Italy

The "Dolomites" covers a big area. Where do you want to go? You can get just about anywhere with trains and buses.
You don't "use the Eurail". Eurail is just a type of pass that non-European tourists, newbies mostly, buy to ride the rails. The other 95% of the people who use trains just buy train tickets.

The Dolomite area does not have nearly the density of rail tracks for trains that Switzerland has, mostly in a couple of valleys. To get into the real mountains, you have to get off the train and take a bus. They are very well connected to the train lines. So you need to know where you want to go, since the trains only get you so far.

Start with www.bahn.com to find out how to get from Graechen to the closest rail station where you can get a bus to where you want to go. An excellent site with all the transport schedules in Alto Adige/Sued Tirol (where most of the Dolomites are) is
https://www.sii.bz.it/en

Car? Eurail?? NOoooooooooo!

In actual fact the Swiss transport system will get you a toe hold into the region. Use the sbb website and plot a route to Mals/ Malles. This is the first town in the Südtirol area of Italy and Swiss Post buses run there at regular intervals via the Ofen Pass. If you have a Swiss Half price card or GA you are only few km left to pay full price for. SII (as linked) are the transport authority for Südtirol and they offer the magnificant Mobilcard http://www.mobilcard.info/en/mobilcard.asp - prices at the bottom, yes it does say €28 for a whole 7 days - Swiss prices end at Mals!!! Note the 7 days can start any day of the week. So if you stayed a couple of weeks you'd buy a second card on day 8. Available from all SII tickets machines, at Mals these are on the station and I think in some doorways of the trains. Valid all regional trains (not EC or Frecce Argente highspeed) as far as Austrian border and as far south as Trento (which is actually out of their area). Also all buses, city and regional buses within Südtirol - actually you can also go out of area on a bus funded by SII (usually silver with bright green ends) if it crosses into another province although they usually connect in a parking area near the boundary. For instance, the Toblach- Cortina bus is run by SII (but the road is currently blocked by an avalanche). There is also a Musemmobil card which adds museum entries for €6 per 7 days.

If the Swiss internal journey seems a bit long you can travel from Brig via Milano and Vernona to Bozen which is the capital of Südtirol. Or via Zurich and Innsbruck to the border at Brenner. From Innsbruck use local train to Brenner and buy Mobilcard there OR use EC train to Bozen and then buy the card.

The eastern end of the Dolmites drops down into Veneto region which has different travel options, and then into Austria and Slovenia. But you would be going a long way if you got that far and maybe 2 weeks wouldn't be enough.

You first have to get to Brig station; I understand this is best done by a bus to St. Niklaus, then train to Brig (with a possible change in Visp). Then you board an international train to Milan, then to Verona, then to Bolzano. From Bolzano, it depends on where exactly you want to go in the Dolomites. It is a bit complicated but not impossible, especially if you manage to keep luggage to a minimum.

Itineraries through Switzerland are possible but look to me very slow and with even more complex changes. You cannot really rent a car as dropping it in a different country is very expensive.

We are use to the Swiss railways but they do not go that far. We have not used the Eurail

Nor has anybody else ever travelled by Eurail, because they are not a rail company! They are a travel agent and ticket reseller operating mostly in North America.
Trains in Switzerland are operated by a number of companies, chiefly SBB (Swiss Federal Railways). Website: https://www.sbb.ch/en/
The corresponding railway company in Italy is Trenitalia: http://www.trenitalia.com/tcom-en

If you decide where in the Dolomites you wish to go (which town), and it has a station, the SBB website should be able to plot the quickest route from Grächen to there.

I forgot to mention. If you are travelling with laptop or tablet in Südtirol download the full regional timetable book to it http://www.sii.bz.it/sites/default/files/kursbuchweb.pdf there is a map of bus routes so get an idea of where you can go, the online journey planner assumes you know that so for a beginner in the area is a bit frustrating. Table 100 is mainline trains - Mobilcard cannot be used on those marked EC or FA, all the others and those on the other rail routes are OK.

For you photos, almost everywhere is nice - and it's on Streetview so you get an idea. East of the main Brenner route railway/ autostrada you get huge rock formations that look like bundles of pillars stuck together, this is very Dolomite.

Many mountain pass bus routes close from early October, the roads themselves can be closed without warning at any time after that

Thank you to those who helped with info. Some very good points fromSam and SteveB, I thank you.

And to those who decided to lecture rather than read my statement and questions, you need to find something better to do with your life. As I said I am very familiar with the SBB. But it is a long way to go through that system into another. And Eurail is a legitimate pass. It just combines all countries.

My main question was about driving not SBB. Just wondering if that would be more fun than sitting on a train for that extended period of time and making a lot of train changes, bus etc. I like to stop and take pictures and explore along the way. You can't do that with the trains.

But it is a long way to go through that system into another. And Eurail is a legitimate pass. It just combines all countries.

My main question was about driving not SBB. Just wondering if that would be more fun than sitting on a train for that extended period of time and making a lot of train changes, bus etc. I like to stop and take pictures and explore along the way. You can't do that with the trains.

"But it is a long way to go through that system into another."
You can use sbb.ch to map your itinerary even outside Switzerland if you'd like. sbb.ch doesn't cease to function at the border. I don't know what you're trying to say/ask with that phrase?

"...wondering if that would be more fun than sitting on a train"
The fun of travel is mostly with the rider. You would probably be the best person to answer your question.

"Stopping to take pictures along the way"
So - here it seems you have answered your own question.

Your attitude comes over a little harsh for one asking assistance.

I think if you are a photographer (as am I) than you must drive. A photographer without a car in the Dolomites is going to be fairly limited on how much they can see or do as the region is served by public transit but likely not on a photographers clock so you are stuck to the bus schedule. Yeah a multiple week car rental is expensive but so is much of the things we do as photographers.

The worry is not the route to take as you have 2 good options but where will the car rental start and end from.
Picking up a car in Switzerland and dropping off in Italy will cost a fortune so you have to figure into your plan a R/T route for dropping the car off in the same country.

By the looks of it you have 2 route choices, I think the longer more Northernly route would be far more scenic.
Option 1 would be to go south to Milan and over on the Italian highways to Verona before heading north to the Dolomites. This would be faster but all highway so not really very scenic and the places to stop in would be cities.
This route could probably be driven in 1 day though.

Option 2 would be to drive north and then east through Switzerland or through Austria before reaching the Dolomites.
This is a longer route due to slower roads which will definitely be more scenic. I think you would stop for pictures along the way so would want to plan an overnight either in North Eastern Switzerland or just over the Austria border or maybe in Liechtenstein if you want to be different.

The 3rd option would be to use the train to get to Austria fairly easily and then drive from there in a rental car which post Dolomites would be much closer to return depending on the rest of your trip and where you fly into and out of. Or train to Zurich and rent there if you are using Zurich as an airport hub.

I imagine each route will have tolls involved (Austria has a highway vignette you would probably need though not certain since not on a major highway) ; around the Dolomites Italy has few tolls but there are plenty around Milan and Verona.

Lastly, there is a European lease option through some cars which may be worth looking into given how long you may have a car for, those leases may allow pickup and drop off in different countries without the huge added fee a rental car would have.

As I already wrote, it is very expensive to rent a car in a country (Switzerland) and turn it back in another one (Italy). Somebody will have to drive it back to its original country and you will have to pay for it.

After that, even assuming you have your own car or cost is not an issue, just play yourself with the sbb website (for train travel) and google maps (for car travel, remember that travel times by car are usually underestimated).

If you choose Bolzano as your gateway to Dolomites, both will suggest my itinerary through Milan and Verona (not really scenic and not worth the extra expense for the drive).

If you choose Malles as your gateway, as it was suggested - but remember that Malles is in a very western position, and still one/two hours by train or driving from Dolomites - you get itineraries through Switzerland. By train is something like 7 hours with a lot of changes and a couple of sections by bus. By car is still a little less than 7 hours (this means true travel time is something like 8 hours), with several major passes to drive. In my opinion, barely possible in a single day and very tiring to the driver; impossible in a single day in bad weather; could be considered if stopping in one or two places in the middle (a possible idea even if going by train:)

Using www.sbb.ch/en, you can get from Graechen to Bolzano/Bozen in 7 hours and 3 changes. Bus to St Niklaus, train to Brig, train to Verona, then train to Bolzano. The price is 123 CHF pp. If you have a Swiss Half Fare, it is 101.50 CHF.

If you intend to be in Switzerland for 2 months, you should probably think about getting an annual Half Fare Card for 185 CHF.

As far as the option of renting a car, that would allow you to go more directly over the mountains with lots of photo ops along the way, and not be tied to the bus schedules in the Dolomites. The trains go mostly in the Italian low country, from north of Milan to Verona.