Hi, I've done my fair share of traveling and I met a lot of great people who were backpacking. But I'll let you know now that you might actually want to consider buying plane tickets in advance to each country you want to visit. It can actually end up being cheaper than buying at the time of, whether you rent a car or take a train etc. Some airlines offer youth tickets if you are 25 or younger. It's better to book through the airline directly so if there are changes you can handle it with them.
What I did was planned 4 months and had tickets already purchased. And between my time in one country, I'd opt for trains or buses etc to other areas or close by countries. If you head to Copenhagen you can actually get a cheap bus to Sweden (Malmo) and that's awesome. Two birds with one stone.
If you end up in Florence there is "florence for fun" and it's for people attending schools but usually if you sign up no matter what you can get really good deals. You could even ditch the group once you get to your destination--they have Prague/Munich trips at a bargain price.
If you do choose a train option, Euro rail has some good options that are cheap over all. But when I took a train in Italy all over, to different areas, it cost me about the same price of a plane ticket that would be worth $600. So consider the EuroRail pass. Meeting locals is best so they can give you rides ;)
You can find a lot of cheap transportation all over though. Do local buses etc. Avoid taxis unless you must.
In Asia, I'm from there, you WILL be asked to pay more if you are obvious you don't know the cost of anything etc. You can haggle a lot in Asia and India on fares.
My tip is to ask before getting in a cab or bike etc.
If you travel with friends, try Airbnb. There are a lot of cheap options if you split between the three or two of you. Some hostels are well priced, but I've found most are 18 euro a night for the cheapest. And if there's two of you, some hosts on airbnb offer a guest room for $30-50 a night anyway so it's in a way better if you want more privacy or relaxing time. Hostels are good to meet various people but some hostels are regrettable.
If you're in spain, in barcelona, try bemar hostels.
If you go to Santorini in Greece, try Oia Youth Hostel. <--Absolute favorite. Easy to get to, nice location, beautiful all around. Sometimes the bathrooms have lines but nice people and staff is very friendly.
I've been to 17 countries so if you have questions let me know.
Sounds like a lot of fun though and whether you choose to get plane tickets or not, make sure to come up with a route that will be best for you to travel within good time.