1. Where to go? 
Depending on our canoing concept, please post your suggestions in this matter. Preferably not so common places, standing aside from main courses - beautiful and wild. Of course visiting URW fans would also be great option!
I think 'where to go' depends on timing and your choice of equipment. Like, would you prefer river routes with rapids, or would you prefer criss-crossing the lake networks? If you have a canoe which is better suited for lakes, then shooting rapids isn't necessarily a wise option. And if you don't have a lot of earlier experience, going out to the sea might not be the safest option. In any case, the the options are plenty.
For river routes May or early June are usually best, as the water levels are at their highest after the snow has melted. Later on in the summer some river routes might become so shallow that they are uncomfortable to navigate.
In addition to great suggestions by Tervaskanto, visiting my place is always an option. I'm located roughly 100 km north from Tampere, and there are several canoeing options nearby. For example
Haukkajoki route starts from Helvetinjärvi national park. I'm not exactly sure but to me it seems that the Haukkajoki route could take you all the way down to the city of Tampere, in case you plan to switch locations by public transport.
2. I also thought about buying kayak or canoe directly in our start area and leave it or sell in the end. Maybe you could help - is it expensive? Is there some internet sites where I could contact with potential seller?
I'd guess there also are canoe rental services, and some of them are fine with starting at point A and collecting the canoe from location B. But I'm not so familiar with them so on top of my head I can't recommend any webpage listing rental options.
For second-hand equipment
here is a link listing 'for sale', 'for rental', 'to give away' options. At the moment second hand canoes seems to sell from 500 € - 2000 €. If you decide to try this option, I think I could personally offer you some help with selling away the canoe once you are done with your trip.
But, of course, a pneumatic kayak would enalbe you to see various locations using the public transport. If you'd buy a second hand canoe, you either need tranportation help from local UrW players, or you are stuck in one water system (which won't be a problem, as all the major lake networks are big enough to paddle around for the whole summer)
3. I plan to fish but I am totally green - maybe you, more experience forumers have some advice for a novice? 
Basic equipment, tips and tricks... I also heard that you could fish with the rod in Finland without license, is that true?
Yes, a simple fishing rod with hook only and no reel - that you can use without a license. With a little bit of luck one can catch enough perches worth a meal.
My tip would be to consider a foldable / expandable portable
fish trap. In the evening when you start to settle to camp for the night, you expand the fish trap and set it somewhere nearby. In the morning you go check the trap, fold it and pack it in the canoe for the day's voyage. That, of course, would mean more stuff to transport in your canoe, but gives the option to catch fish while you sleep.
Using a fish trap would require paying the national-level fishing license (they sell for 15€ / week. If you pay that, it would also allow you to drag a lure while you paddle around. Another nice way to possibly catch a pike while you go.) To be precise, a fish trap would also require a local-level license depening on the waters where you are. But my personal opinion is that if you are switching places every night, and you aren't highly experienced fisher then you catches are likely to stay so small that paying a local fee isn't that necessary.
4. Buying food (and of course alcohol;) ) . Let's suppose that I won't catch a single fish - Some advices regarding supplying?
Regarding alcohol, anything stronger than 5.5 % is only sold at special liquor stores, which can't be found in every small village. So if you want to have that bottle of dark rum to warm you up in one of those rainy evenings, plan to buy it beforehand while you are in a city or a town with liquor store (they are called "Alko"). Otherwise I'd guess it is the basics of camping food. Dry and canned stuff. Things are a bit easier if you have a portable stove - that way you don't have to start a fire every single time you'd like to have a mug of tea. (Ah, especially in May / early summer most of the water is safe to drink. And becomes safer once boiled. So packing a lot of drinking water isn't absolutely necessary. You can boil lake water every night and store it for the next day.)
It never hurts to have a bar of dark chocolate for quick energy if in situation where you need it.
5. Traveling in Finland. Quickest, cheapest way of traveling - by bus, train or maybe car sharing?
Trains are the quickest and often the most comfortable way of travelling. Buses can be either a lot of cheaper or more expensive. (Like, where an ordinary train ticket would be 30 € / adult, with a luck you can buy a bus ticket for the same route for mere 2.50 €.) I think train and bus operators all have such a pricing scheme that the earlier you buy your tickets the cheaper the prices.
For buses see
matkahuolto.
For trains see
VR.
I know there are car sharing services out there but that's pretty much all I know about them. That they exists.
For now this all that come to my mind, probably I would have more questions later.
So, if you are planning for summer 2019, feel free to start a separate topic when you get into more detailed planning.
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Trip to Finland
Topic: Trip to Finland (Read 22985 times)