Needs suggestion for my Europe tour itinerary

Hmmm... Original idea to take the train from Rome to Norway. Not the cheapest way, nor the fastest, but at least the most interesting, if not the most pleasant. A rough idea would be: Bologna - Innsbruck - Nuremberg - Berlin - Hamburg - Kopenhagen - Göteborg - Oslo. But there are dozens of other possibilities, with more or less stops, different itineraries and detours, depending on your time and interests.
Get this pan-European railplanner to play with possible itineraries and the excellent website of Man in Seat 61 for practical information on European train travelling.

Forget Norway and forget Rome. Spend 3 nights in each to give you just a taste. Travel by train from north to south, following the last days of summer.

October
06. Arrive London
09. Paris
12. Nimes, Provence
15. Barcelona
18. Madrid
21. Seville
24. Depart Seville

If you are searching for a journey by train from southernmost to northernmost Europe, through 8 countries.

Algeciras, Spain (latitude 36' N)
Madrid, Spain
Barcelona, Spain
Paris, France
Brussels, Belgium
Amsterdam, Netherlands
Hamburg, Germany
Copenhagen, Denmark
Stockholm, Sweden
Narvik, Norway (latitude 68' N)

"Best" can often go hand-in-hand with "fastest", but "cheapest" rarely goes with the other two.

places to visit are London, Paris, Rome, Spain then Norway. Date travel will be on Oct. 6 - 24, 2017 (18 days).

You have proposed a very ambitious trip! Are you sure you want to do all this in the time you have? First, you don't have 18 days. You land in Europe the day after you leave Orlando. The last day is taken up with getting to the airport to get back to Orlando. The day of arrival is often a jet-lagged haze, so you shouldn't plan anything major on this day. You have to allow a half day for changing locations that are close together, and a full day for more distant places (like those involving a flight between them). Finally, two nights in a place equals a full day; one night in a place means less than a full day. It's easiest to count nights when planning.

So, if you fly out October 6 and fly back on October 24, you have only 18 nights. You have proposed three big cities and two countries in that time. If you insist on this, you will be spending a lot of your trip in transit, and packing and unpacking, rather than actually seeing places.

Which is the best way to start the trip?

You will have to check prices to and from various cities for you dates. There is no rhyme or reason any more in airfares. Be sure to look for multi-city fares, into one city and out of another (for instance, Orlando to London and Oslo to Orlando). This is usually much cheaper than two one-ways.

Then planning to go to Norway from Rome by train. Which is the best/fastest/cheapest way to go to Norway?

As said above, best and fastest and cheapest are three different things. Best is in the eye of the traveler. Fastest and sometimes cheapest is flying. Bus is very cheap but takes a long time.

Is it from London or from Rome? Also, What train we need to ride?

To find train routes within Europe, use the Bahn (German Rail) website http://reiseauskunft.bahn.de/bin/query.exe/en, following Rick's tutorial: http://www.ricksteves.com/travel-tips/transportation/trains/online-schedules. This will only have prices for trains that start or end in Germany.

For all rail matters, the rail guru The Man In Seat 61 is great. For instance, here's his directions on getting from London to Norway by train: http://seat61.com/Finland.htm. You see right up to that it takes 3 days and 2 nights. Do you really want to spend this much time out of your trip in transit? I wouldn't, but others would - your call.

How many days that we need to spend and able to see most of the tourist destinations in each country?

How long is a piece of string? How long to spend will depend on exactly what you want to see. For me, a MINIMUM would be 3-4 nights each in London, Paris, and Rome. Spain is huge and you could easily spend all 18 of your nights there and still only see famous places (Barcelona and around, Basque Country, Madrid and around, Andalucia). Norway needs 1-2 nights each for Bergen and Oslo, plus any time you're spending on the natural scenery (which takes some time to get to).

If you want Rick Steves' suggested highlights of each place, click on each country in this link: https://www.ricksteves.com/europe. Remember these are only his preferences.

"What hotel to stay in Paris, Rome, Spain that is very economical? "

For very economical places to stay, look at EuroCheapo http://www.eurocheapo.com/, HostelWorld http://www.hostelworld.com/, or HostelBookers http://www.hostelbookers.com/.

On our way to Norway via train, which places/countries that we must stop to see tourist attractions or do we need to stay at least 1 night on those must see countries?

Again, up to you to decide "must sees."

We have luggages with us is there a place where we can store or place our luggages while we are touring around the place? How about in the train stations do they have a concierge area?

Some train stations have "left luggage," which is the name for where you can store bags. Some have lockers. Some have neither.

Add my voice to those who are gently suggesting going back to the drawing board and narrowing down your itinerary to something that will fit into 16 days. (Yes, the OP said 18, but first and last days won't count for much.)

No one can do this but marirosb, and it is hard work, but necessary to plan a successful trip. Really good trips succeed by minimizing time spend checking on and out of rooms and waiting for planes and trains. Guidebooks are indispensable planning resources that can help you decide what your true priorities are and how they might fit together.

Not a feasible or advisable trip for 18 days. Have you ever been to Europe before?

Serious trimming of the proposed itinerary is needed, else your trip is going to consist of not much more than riding on trains and buses. Which would be a shame. You seriously need to read some guide books. Start with the country that is your number #1 priority (perhaps Norway?). Read about that country. Couldn't you spend all your time there?

You will not see more or have a better trip by stringing together a long list of destinations stretching across 5 widely scattered countries. Excluding scenery seen through the bus or train window, you will actually have time to see very, very little, because you will always be checking into and out of hotels, going to and from bus and train stations, and sitting on moving vehicles. This is the formula for a rather expensive trip that will not be very memorable--at least not in a good way.

Thank you so much for all your inputs/suggestions/advices. I will take considerations all your recommendations.

Hi,

Fly to Norway, see Oslo, take the ferry Oslo to Kiel (Ger), then the trains until you get to Rome. I would seriously consider dropping Spain. That makes the trip logistically more feasible. Or, you can start the trip in London, end by flying out of Oslo. Still use the ferry between Kiel and Oslo.