Public transport isn't really a problem in Europe (compared to the US) Unfortunately if you aren't a european (and sorry I can't see if you are - but I am guessing you're from North America??) then you can't take advantage of Interail but even so, if you plan well and can book big journeys in advance you can make some substantial savings and still cover alot of ground.
Also you may be able to consider cheap 'internal' flights if you are willing to shop around and consider non-major airports as destinations. That may not be a problem as though you should definitely see cities like Paris if you can they will cost you a lot compared to other areas. London, especially, is very expensive.
Again, if you are from the US this is the worst time in 20 years for the currency exchange but with careful planning you should still be able to have a good time on a budget. As the previous poster said Hostels are great, very cheap and lots of websites exist to recommend the ones to consider and which to stay away from. Even big cities have hostels and you can always find places to eat that don't cost the earth if you look around.
Lots of websites will exist for people who did things very similar to what you have planned so take note of those.
Safety tip - I was very much like you in terms of confidence etc. But it's important that (if you find yourself in an area you'd rather not be in) you treat that just like you would at home. Look confident. Look like you know where you are going. Don't look at maps etc while in such an area. I'm not trying to worry you here - I doubt you'll end up in such a place, but if you did try not to look like a tourist.
Food tip - in the UK, we don't get huge portions and you pay as much in £ as you would in $. And - as a rule - our service staff aren't usually in a good mood or working for tips so... don't be offended if you don't get the chatter you might in the US.
Food tip 2 - esp in France/Spain etc, markets are your friend. You can buy wonderful bread, cheese, fruit etc at a market and it will be both cheap and better quality than you'd get in a store. It's simple food - but if you can wait for breakfast until the markets open, you can get two daily meals really cheap. (As a rule this doesn't always apply in the UK...)
Food tip 3 - Don't be afraid to try the local food! If you are in a seafood area (and like it) then that's probably going to be the cheapest stuff you can find, give it a try.
Food tip 4 - Ween yourself off coffee before you go. Going into coffee shops is fatal on a budget. You have the cost of the drug you crave, then the lovely looking cakes etc... You can do without it, trust me on this. If you can't buy a jar of coffee when you arrive and make it in the hostel or hotel.
Food tip 5 - Always take a print out of common food stuffs in the host language. A menu that also is in English is likely to be a tourist place and thus more expensive. At least with a print out you can figure out what you are ordering!
Langauge tip - Make the effort. If you try to speak a language and fail, you'll be received much more sympathetically than if you just point harder and speak s l o w e r & LOUDER. The French (sorry to sterotype) are reputed to avoid being helpful to foreigners. Not so... But they definitely respond better if you at least try to get it.
Travel tip - Don't be surprised if you end up buying public transport tickets in "strange places." Just for example in Italy some types of subway tickets either come from machines (often broken) or over the counter in a tobaccanist! Just because that's the way it always used to be I guess.
Money tip - Set a daily budget and (assuming you can lock the rest of your money away) only go out of the door each day with 1.5 times the budget in your pocket. (The extra .5 is just in case you run into something unexpected.) Even with travellers cheques (excellent idea as Tim said) don't put all your eggs in one basket.
Insurance - get it. Make sure that either your existing health insurance covers it or get some trip insurance that includes health. It's not like the US, they aren't going to charge you $10000000 to fly you home and deal with a broken leg but there are substantial costs and for the sake of a small premium you can rest easy on thuis score. Hopefully it will be wasted money.
Planning Tip - Check what is happening in that country/city at the time you are there. You don't - for example - want to be trying to get a room in Paris the weekend that the Tour de France rolls into town. Also Paris pretty much empties in August when the French go on holiday. This leaves the chain stores open but not so many of the local ones. That pushes your costs up.
Finally (assuming you are from the US) if in doubt... pretend to be a Canadian

(Except in France as then they will wonder why you can't speak French!)
Katharyn