Hello srajani786,
Welcome to the forum! I'll do my best to get this thread on its feet.
>Do you think we will have enough time to visit any other cities
For three weeks, your itinerary is plenty. As of now, you have to time to visit each city without feeling rushed, including cushion time to travel to and from each destination. If you want to visit more cities and still save money, visit places that are closer together. Less distance covered = less money spent.
This
Railway Map (PDF) from Eurail may help you a bit.
> Cheapest transportation
That depends on 1) when you leave 2) when you buy the tickets 3) where you are coming from/going to 4) how fast you want to get there
1) Transportation is more expensive in the peak season. I assume you'll be travelling during the summer? That means peak prices, as far as airfare is concerned.
2) Tickets cost less when you buy them in advance. For example, the train from Paris to Amsterdam costs 25E - 120E O/W, depending on when you buy the tickets. As soon as you are sure of dates, purchase tickets online to reap the benefits of early bird discounts.
3) Popular routes will generally cost less, provided you purchase tickets in advance.
4) Time is of the essence, especially when a trip involves crossing large distances. Compare these two options:
Elipsos Train Hotel - Barcelona Franca to Paris Austerlitz - from 69E ~$102 USD - 11h58
Vueling Airlines - Barcelona to Paris CDG - from $77 USD - 1h50
The train is attractive because it takes you from city center to city center, but it takes six times as long as the plane. With flying, you don't have to worry about sleeping on a night train (some can; others can barely doze off while in motion) or losing half a day of travel time. If you have 12 hours to kill and are able to sleep on trains, consider Elipsos. Otherwise, you'll have an easier time flying. Be sure to arrange city-airport transportation beforehand.
Semi-rule of thumb: If the train and plane costs are close AND the train ride is less than 5 hours, take the train.
Transportation websitesLa SNCF - for trains involving France
Thalys - Paris-Brussels-Amsterdam train line
Die Bahn - complete listing of European train timetables - can only purchase tickets for journeys involving Germany
Sidestep - useful for finding flights - this is my favorite, but there are many others.
>Saving Money in Europe
1. Don't eat out often. Be like the locals and buy groceries.
2. Stay in hostels.
3. Keep an eye out for youth (under 26) and student discounts, free nights at museums, free attractions, and clubs with no cover charges.
4. Wear a moneybelt under your clothes to discourage pickpocketing.
Hope that helps.