I would only see one city given that you only have 5 days.
Taxi from Budapest airport www.fotaxi.hu
Flights Budapest to Rome www.wizzair.com
Rome airport bus www.sitbusshuttle.com
Currency exchange www.xe.com
It sounds like the OP has already made up his/her mind. I would fly into Budapest first then fly into Rome. I believe there are some low cost airlines that service these cities. Then from Rome to Venice, I would take the train versus flying. It's only 3 hours+ via train, and if you factor the amount of time you need to get to and from the airport, the time difference is not material. Besides the Termini (Rome) and Santa Lucia (Venice) stations are conveniently located in their respective cities.
Personally, I wouldn't do this itinerary, but you must have your reasons. There are plenty of things to see in each of these cities for an entire week.
There's a typo from @Nurioopta. The other train line is www.italotreno.it. It does seem you'll spend half the time in transit...
Thanks everyone for your comments
Arriving London early morning for 2 nights from syd 1st time there ideas on what to do
-:- Message from TripAdvisor staff -:-
This topic has been closed to new posts due to inactivity. We hope you'll join the conversation by posting to an open topic or starting a new one.
To review the TripAdvisor Forums Posting Guidelines, please follow this link: http://www.tripadvisor.com/pages/forums_posting_guidelines.html
We remove posts that do not follow our posting guidelines, and we reserve the right to remove any post for any reason.