First of all, I have to mention that there is a huge difference between flight costs. Normally earlier you book your trip the better price you can get, at least with low cost companies. I like looking my flights thru Norwegian airlines, Ryanair and momondo. Even if I find a flights thru some page like momondo or edreams, I usually book the trip thru the original page of the airline company I'm flying with. It's same price and easier for booking extra luggage or handle any problems.
Luggage: You can fit pretty much things on hand luggage. Just make sure your bag is light, so no extra weight is lost on that. It's easier to get to airport with less things with you and also it's cheaper with some companies if you don't need a check in any baggage. When I travel with BMX bike I take it to airplane in icehockey bag. There I can fit all my riding gear as well as many clothing. Helmet I have always had as an extra hand luggage without any problems. And inside the helmet you can ones again fit much things..
Many airline comanies, as Norwegian for examble, are teeling on their website that as long as your luggage is the size and weight they are accepting, they don't care what you have inside your luggage. So you can take your bike without any problems. But if I'm asked what do I have in my luggage I simply answer sport equipments. For further questions I say bike parts. And never have had a single problem. When you are taking your luggage to airport and you have a guess that it might be too heavy, first put the most heaviest one on the line. So you are still able to move things from bag to bag. You are allowed to have as many clothes as you want when you walk in the plane. And of course ones you are in the plane you are allowed to take off all the extra clothes that you have wrapped around you. Big pockets in a jacket are my final call to get away from paying for extra kilos. And what comes to handluggage you can take surprisingly many things in to your hand luggage. If you are not sure about something just call to the airport before and ask them. I have took with me for examble computer screen, bike parts (hand breaks, cables, tires, camera stands made by metal).
Travelling with bike is not only weight. I have even rode on my BMX to the airport. But the bike in to the parts on the airport lobby. Send the bike to the plane and when arrive take bike again and keep on riding in other country. So I kinda like travelling with bike ;)
On budget you of course have your own snacks. Sometimes I have them even on the flights where is served meals. Your own snacks are just much more healthy and costs hardly nothing. No need to be ashamed to pull out your own lunch box. There is even many times that passenger seated next to me is looking jealous and telling the wife that next time he wants have own lunch box. Drinks had made to be little challenge but it's pretty easy. I have my own drink bottle with activated carbon filter so I can take water from any place. On top of that I have my own 100ml drink bottles (4) that I can take thru security check.
If you need a car you need it. You got to rent it, but judiciously. There is cheap car rents, there is also hidden costs. Extra insurance is pretty much up to you if you need it or not. Normally in Europe every car has insurance in case of crash. So for example in Spain the extra insurance is only for windows and tires and in case of robbery. Main thing is to check the car in good light and make the renting company to write all the smallest scratches down. There as been cars that has been rented to me with "clean papers" and those cars had have huge dents!
When you need something more special asking tips from locals isn't a stupid idea. For example many countries has nicely working shared rides pages on web. Where you can get much cheaper ride from city to city than it would be on public transportation.. Or locals might now which bus companies are the best. More extra thing I usually have spoon and army knife with me. It's much more easy to find cheap snacks when you have those. Sleeping bag and towel are pretty good when you don't know what's gonna happen. And when I travel by car having tent and blow up mattress has saved my ass many times.
So calculation for simple trip to Finland:
To have only hand luggage: saving 20e/one way
Parking my car for a week, saved 30e
In Finland used public transportation to/from airport, saved around 10e/way
Own luch box, saved 6e/way
Water bottle, saved 3e/way
Own juice, saved 2e/way
When my flight to Finland costs around 70-150e round-trip, and I'm saving 100e from extra costs. I think that's worth it.
Many people when they get on travel they are just spending lots and lots. I don't judge anyone, but if you want to travel more my advice is to enjoy the locals habits, enjoy the nature and get to know the people in the country you travel to. Tourist attractions are nothing what I'm keen on. Most important advice I have is that don't make budget travelling by making other pay for you. And be happy positive person and everything is gonna be ok. (For example one time in Azores island. I didn't have place to sleep and my surf teacher heard about that. He told us that he has one extra house that we could stay. He gave us the keys and we slept there. He's name was Dario, I still remember that because it was a huge thing for me. Probably not big deal for him.)