I think most people access the Dolomites by taking a train to Bolzano then hopping on one of the regional buses. Traveling from Venice, there would be a change of trains in Verona.
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Trenitalia and Italotreno operate trains in Italy. Italo covers only major destinations; it doesn't go to Bolzano. Trenitalia requires you to use the Italian names for the rail stations: Venezia S. Lucia for Venice and Roma Termini (or Roma Tiburtina) or Roma Tutte Le Statione (for "all stations") for Rome.
Seat61 has fabulous background information on the Italian trains. The amount of info can be overwhelming (though you can skip over all the stuff about taking the train to Italy from other countries), but reading through it a few times may be very helpful. You'll be able to save quite a bit of money by buying your rail tickets well in advance (a few months ahead, usually), but the cheapest tickets will be non-refundable/non-changeable, so that's something you won't want to do until you have your timing locked down. Tickets for your travel period won't be on sale for quite some time, anyway.
I haven't stayed in a youth hostel since the 1970s, but I know some of them now have private rooms, even with their own bathrooms. If you need to pinch pennies, I think that would be a better way to do it than having to buy your food at supermarkets or skipping sights you'd really like to see because of their entry fees. If money isn't too tight, for a honeymoon I'd opt for hotels instead. They needn't be expensive as long as you don't mind rooms (and bathrooms) that are quite small by US standards. You can explore options on websites like booking.com, but I suspect the prices you see now may be lower than what you'll encounter late next year or in 2022, when the demand for travel is normalizing. I find the reviews on booking.com very helpful; you can't leave a review there unless you've actually stayed at the hotel.
I loved my time in the Dolomites, but I was there in the summer. It's a beautiful area with not much to do indoors, so do some weather research if you're thinking of a fall, winter or spring trip. I like to use the website timeanddate.com, because it has actual, day-by-day, historical weather data going back about 10 years. I find that a lot more useful than monthly averages that hide the extremes.
Keep in mind that altitude has a huge impact on temperature. I've linked to the May 2020 weather stats at Ortisei, a mountain town many of us like (though there are many other options). Bolzano is a very nice city with a lovely historic area, but it is down in the valley. It will be a lot warmer than the mountain villages; Bolzano can be unpleasantly hot in the summer, for example. We normally recommend that folks who want to see the Dolomites stay up in the mountains rather than down in one of the valley towns like Bolzano, but the Museum of Archaeology in Bolzano has an excellent exhibition built around the Iceman; it's very much worth visiting.
In case you are not already aware, the Dolomites have an extremely strong Austrian influence. It affects the local food, architecture and language. So if you're able to visit that area, it will be a bit like you've been able to include Austria in your trip.