Train from Venice to Budapest with bicycles

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Can someone please advise us how to get from Venice to Budapest by train with our bicycles?  We had understood there was an overnight train via Vienna, and that we might need to book an extra berth in compartment (ie, the full compartment) to keep our bikes with us - but now cannot find information and where/how to book. Help please!

Can someone please advise us how to get from Venice to Budapest by train with our bicycles?

Well, there are two major rail options for journeys from Venice to Budapest:
a) overnight trains from Venice to Vienna (currently 20:57-8:08 or from Venice-Mestre 1:30-8:30) and then change in Vienna for train to Budapest, nearest is at 9:48 (Wien West) or 10:03 (Wien Meidling), arr. to Budapest-Keleti at 12:49. However neither of overnight trains have special compartment for bicycles, it is only available in Venice to Munich carriages (depart from Venice with ones to Vienna, but get reattached at Salzburg). So you can either buy out the whole compartment (although maybe better to email Italian Railways first and actually ask whether you need to do so) or do a trick with changing trains in Salzburg, thus from Venice to Salzburg you bicycles will travel in Venice to Munich carriage, while from Salzburg to Vienna you continue by Austrian train that has place for transporting bicycles
b) travel via Slovenia-Croatia. Daytime train from Ljubljana to Budapest via Zagreb has a coach with special place for bicycles and bonus is that ticket from Ljubljana to Budapest is pretty cheap (compared to the option via Vienna), however the problem here is that train to Budapest leaves Ljubljana really early 6:35am, so in order to catch it you'll have to spend night in Slovenia. Otherwise it's pretty easy to get from Venice to Ljubljana by train - either via Trieste-Sezana or Gorizia-Nova Gorica

...but now cannot find information and where/how to book.

You can book tickets from Venice to Vienna (or Salzburg) via official Italian railways website - trenitalia.com, tickets within Austria or from Austria to Budapest can be booked via official website of Austrian Railways - oebb.at (here we also have a step by step guide on how to buy trains tickets in Austria via OeBB website - http://europetrainsguide.com/Advice/How-to-Buy-Train-Tickets/Austria-OeBB.html )