I'd like to know if someone can give some tips for the best route.
That should include: Paris, Amsterdam, Italy, Barcelona, Berlin, Zurich and maybe Austria, Prague...
Any ideas about the best transport to be used around this cities?
Answer on I'm going to travel together with my 2 brothers around europe for 21 days.?
Seems like you're thinking of travelling capital cities, so assuming that, your trip if possible will include Paris, Amsterdam, Rome, Barcelona, Berlin, Zurich, Vienna and Prague.
A possible route, without backtracking, would be :
Paris - Barcelona - Rome - Zurich - Vienna - Prague - Berlin - Amsterdam.
To be honest, 7-8 cities in 21 days is really not feasible, unless you plan to barely skim the cities by, say, arriving in one, going on hop-on, hop-off buses and take a quick tour, see one or two main sights, and out of the city to the next destination.
I would suggest cutting the number of the cities down (and thus also reducing your transportation costs) to 4, at most 5. Perhaps to something like Paris - Rome - Zurich - Berlin - Amsterdam, or Paris - Rome - Vienna - Berlin - Amsterdam.
To travel between these cities, the most convenient option would be by train. For two cities with longer distance e.g. Paris-Rome, Rome-Vienna, you can consider taking overnight train. This way you'll waste less day time in travel, and you'll save some money by not having to pay an accommodation (hotel/hostel) for the night.
It may be worth it for you to get an Eurail railpass. Given your itinerary, the suitable pass is the global pass and for a pass that allows 10 days travel within 2 months it costs 393 Euro if you're 25 or younger, on a second class train ride. If you're taking overnight train or high speed train, there are supplementary charges.
Research the ticket prices for individual journeys first, and the earlier you get them the chepaer they tend to be. Normally tickets can be bought up to 3 months prior to travelling date. If they add up to be cheaper than the rail pass price (remember to account for supplementary charges where applicable) then don't get the rail pass and just buy the tickets accordingly.
As for travel within these cities, like all good European cities, within the city center area, walking is actually the best option most of the time. And it's free. I walk pretty much everywhere in these cities when I visited them (so good and comfortable walking shoes are important!) ;)
Otherwise, metro/underground, buses and trams are the other ways to travel within the cities. They also use the same ticket between the different systems, which is very handy. Here are some tips on what I'm familiar with the public transports in the cities that I've travelled to multiple times that I know more than just walking around or buying single tickets as I went along.
a) Paris - Tickets can be bought from ticket machines/counters at metro/RER stations, or from the tabac around the city. A carnet of 10 tickets costs 11.40 Euro and each ticket is good for a single journey in Paris central zone. Single ticket is 1.60 Euro each. The one day Mobilis ticket (zone 1-2, for use in the city, 5.80 Euro) is not worth it unless you plan to travel at least 5 journeys or more in a day on the public transport system. Don't get the Paris Visite card - it's a rip off.
There are Velib bicycles everywhere in the city that you can take out for use at minimal costs (as cheap as 1 Euro per day) but I wouldn't really advise first time visitors to be cycling in Paris as the traffic can be quite daunting.
b) Barcelona - Get a T10 ticket at 7.20 Euro - this ticket allows multi person usage for 10 rides. Otherwise there's also unlimited travel tickets, e.g. 2 days ticket for 10 Euro. Getting from one quarter to another may be easier using the public transport, since the city is quite a sprawl between districts.
c) Rome - Get the tickets at the tabacchi. Single ticket is 1 Euro each, and is valid for unlimited travel within 75 minutes following ticket validation. Otherwise the day ticket is 4 Euro, or 4 days ticket is 11 Euro. The ticket is also valid for using the underground/metro. Be careful while you're using the public transport (esp on main tourist bus routes and when it's busy and packed) as pickpockets are in operation and they are very very quick!
d) Vienna - Tickets can be purchased at machines/kiosks at the train stations, or from tabak. Single ticket is 1.70 Euro, a 24-hour ticket is 5.70 Euro, or a three-day pass for 13.60 Euro.
e) Amsterdam - This is one city where people are either walking or cycling. You can rent a bike for 7-10 Euro per day. If you are walking, keep an eye on the cyclists. Goodness know how many times I was nearly ran over by them.