Transportation from Venice Marco Polo airport to Dobbiaco

Best starting point is https://www.rome2rio.com

1 bus + 2 trains to Belluno. Then 1 train + 2 busses to Dobbiaco.

The faster options are probably the following:
First you go from the airport to Cortina with one of these two buses
http://www.atvo.it/allegati/linee/linea_29_dal_17.06.2017_al_31.12.20174100.pdf
http://www.cortinaexpress.it/wp-content/uploads/ORARI-ESTATE-2017-CORTINA-BOLOGNA.pdf
Then at Cortina you'd take this other bus.
http://cortina.dolomiti.org/uploads/Pagine/sad%20dobbiaco-cortina%20estate%202015.pdf
Having said that I would definitely rent a car. First of all because you can get there in half the time, secondly because in a mountainous rural area like that transportation is not a frequent as in urban areas, and a car would give much more flexibility.

Belluno would not be on the way in any case, and it would be a sligh detour. I don't know why you need to drop gear exactly there. Do you have relatives in town?

Use http://www.trenitalia.com/tcom-en to get the route using trains. Take the No 15 bus from Marco Polo Airport to venezia Mestre station. That will give you the route to Belluno.

Then use that site to get you to Calalzo-Pieve di Cadore.

Then a bus from there to Cortina d'Ampezzo, http://dolomitibus.it/files/orari/2017-estivo/extra/Cadore/linea30_E.pdf

Then the 445 bus from Cortina to Dobbiaco. Search Cotina d'Ampezzo Autostazione to Dobbiaco Autostazione.
https://www.sii.bz.it/en/

Your wanting to stop in Belluno on the way makes this a rather tortuous journey.

Two points,

your knowledge of Italian isn't a big problem in northern Italy. Because they mostly speak german! Dobbiaco is the italian name given to the town of Toblach by Mussolini and Co. when they tried to eradicate the original austrian culture. But they failed. https://en.wikipedia.org/wiki/Toblach

second, the road from Cortina to Toblach is currently blocked and the 445 bus is not running "for a few days" - best check again next week. and the road from Belluno to Cortina is also blocked
http://www.tageszeitung.it/2017/08/06/gesperrte-strassen/ - glad I wasn't in the white van"

SteveB
Where did the OP say that s/he speaks German.

The road from Cortina to Dobbiaco was washed away by a sudden storm two days ago. I would expect it to be reopened in a few days.

Dobbiaco has the distinction of having its own railway station, a distinction rare in Dolomites. In order to reach Dobbiaco by train you have to go through Verona - Bolzano - Fortezza (changing trains first time in Verona and second time in either Bolzano or Fortezza). But you do not pass anywhere near Belluno.

But the OP has to stop in Belluno to drop extra gear that is (I presume) too heavy to take on this trek, which I am again assuming, will end in Belluno.

Thank you everyone, I believe Sam's response from Green Bay helped the most. Yes, we would like to drop extra gear in Belluno which is our last stay after walking 8 days. The extra gear consists of items we do not wish to carry during our 8 day trek but will need it for other adventures in Italy. We may look into posting it to Belluno from Venice rather than trying to get there personally.
Chris

Trasbagagli will store your luggage at Venice airport for 7 EUR per day, so that is pretty pricey. It is a little cheaper if you go to the Trasbagagli office at Tronchetto in Venice for 5 EUR per day. If you choose to do this, i would advise you consolidate it into one very big suitcase. You will have to return to Venice to retrieve it, but then you pretty much have to return there from Belluno to get any where else in Italy. And relaxing in Venice for a few days after an 8 day trek is not the worst idea.
http://www.trasbagagli.it/en/pg/view/53/luggage-storage-venice-piazzale-roma-airport

Where did the OP say that s/he speaks German. - s/he didn't - but most of the people of Toblach do. So lack of Italian isn't a problem there.

Yes. But if you don't speak either of the two languages spoken there, what is the difference?
A more sensible comment would have been that English is widely spoken in most tourist areas, therefore not speaking Italian or German, but at least speaking English, would have been sufficient.

Whoa! Don't want to start a ruckus, it's my lack of capabilities using either German or Italian that is the problem. I appreciate all the help I can get.

German as spoken in Dobbiaco is a local sub-dialect of Bavarian. I would not be amazed if people who have studied German in school, or even people from northern Germany, could find it difficult to understand. Not as impossible as the German from Voralberg or Swiss-German, however.