I am debating whether or not to turn in my rental car or train it the rest of the way. How easy will it be to visit Ricks recomended side trips from Budapest using public transit?
Not sure what side trips you have in mind, but the top 10 day trips and overnight trips are all on the rail lines for under $15 a ticket. Cars generally don't serve good purpose in and around Budapest and are expensive to park.
Rick has all the details in his book about getting to his recommended places by public transit. The only one that seems less easy by public transit than by car (I didn't do it so I can't personally comment) is seeing more than one Danube Bend town in one day. Otherwise, I got to Eger, Pecs, Szentendre, and Vac (last one is not in Rick's book, but well worth seeing) by bus and train with no difficulty at all.
Yup, to do all 3 in one day is a bit of a challenge, even by car. My favorite are Szentendre and Esztergom and you can pull that off in one fairly active day with public transport.
2 out of 3 on the Danube bend using public transport. Gives you about 2.5 hours in Szentendre and 2.5 hours in Esztergom. Sort of a minimum for each. But you do get a really cool boat ride back to Budapest.
Train Budapest Nyugati station to Esztergom : 8:20 to 9:44
Bus Esztergom to Szentendre: 12:38 to 14:15
Boat Szentendre to Budapest Batthyány tér 17:00 to 17:50
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