Budget airlines don't fly to the main airports, and are usually further out, so the savings are kind of negated by transport to/from the center.
Fuck Eurail passes too, just buy point to point when you need, will probably cost the same or less.
To save money overall, spend more time east of Germany, check out countries like Czech Republic, Slovakia, Hungary etc
This is true about the budget airlines to an extent, although in some cities there's only one airport so it's not an issue in those places (e.g. there's only one airport in Dublin and the budget airlines serve it)
I suggest planning out your trip in advance. If you book your plane/rail tickets in advance you should be able to find good deals. Also, I would avoid getting one of those flat rate rail tickets. I know people who got them and they always felt under pressure to get back on the train to get value out of it, but without spending enough time in a place to enjoy it.
The biggest budget airline is ryanair. There's also easjyet and others. Don't rely on sites like kayak.com etc to find flights. They don't really cover budget airlines (e.g. I looked up flights from Dublin to Sardinia, and kayak.com suggested it would be €1k and two stopovers, but when I looked at the budget airlines, it was €90 for direct return flights!).
Cities in Eastern Europe are cheaper and lots of them are worth seeing. Places like Paris, London, Dublin etc are much more expensive, but they're still possible to see on a shoestring. I think planning is key, and looking out for bargains. In Ireland, for example, the hotel industry is now fucked and there's more capacity than demand, so lots of places are offering rooms cheap. Shop around and you'll get some good deals