You'd spend all your time dealing with flights and inconveniently located airports. You're left with no flexibility, since so much has to be booked in advance.
Instead, look up how train ticket prices are set -- for example, in the UK a budget backpacker would need to book ahead to get at affordable price, but there's no need in Poland. If you miss a cheap pre-booked train in the UK you need a new ticket, but on most trains in Germany you've only lost the seat reservation (I think).
Buses are often cheaper than trains, but much less comfortable and have poor views along motorways -- only use them if the train is too expensive, or the journey a lot more direct / faster.
You then have the flexibility to spend longer in one city than you planned, or to leave early when the weather is bad and you're bored of museums, or to break journeys in smaller towns and villages, which makes for a more interesting trip than a succession of major cities.
Students with a very tight budget should stick to Eastern Europe, and not only EU states. There are Roman ruins in Albania, vineyards in Romania, modern war history in Bosnia Herzegovinia, and beautiful 19th century buildings in Hungary.