Eurail pass

Rome - Venice - Yes covered, but you have to pay a surcharge (called a reservation fee) as this is a High Speed train.
Venice - Florence - Pisa - Yes, + surcharge as above.
Pisa - Lucerne- Yes, + surcharge as above for the Italian section only.
Lucerne - Paris - Yes, + surcharge as above for the French section only.
Paris - London (Eurostar) - Yes, + Surcharge.

I seriously suggest, before committing to a Eurail Pass, you investigate the costs of buying these tickets directly from the railways running the trains. Italian Railways, French Railways and Eurostar all give big discounts (50%+) for advance purchase tickets. Individual tickets will probably be cheaper.
You need to look up fares on the railway companies' own websites. Resellers usually charge more.

In Luzern, most visitors make trips up mountains or boat trips on the lake. These are not covered by a Eurail pass.

Johny, Have you read the "Man in Seat 61" website? If not read these two pages before buying anything:

A beginner's guide to Train travel in Europe: https://www.seat61.com/Europe-train-travel.htm
A guide to buying cheap European train tickets . . .: https://www.seat61.com/Europe-train-tickets.htm

You might also read this adjacent post about the headaches having to buy those reservations can be.
https://community.ricksteves.com/travel-forum/italy/train-reservations-switzerland-to-italy-how

First, it's great that you came here and asked BEFORE buying the pass. We often see posts from people bought first and only came here after they had problems.

The Man In Seat 61 is a great resource for European trains. Before you do anything else, read his discussion of passes and the issues involved: https://www.seat61.com/Railpass-and-Eurail-pass-guide.htm#railpass-or-point-to-point-tickets

If you can commit to advance purchase, non-refundable tickets, you will save a lot of money over a pass. Use Seat 61's handy guide of where to book each ticket: https://www.seat61.com/Europe-train-tickets.htm

For many of your trips, advance tickets will be cheaper than last minute tickets. You didn't specify your travel dates, so we don't know if you can still snag discounts. If you can't, a pass becomes more cost-effective. However, you still need to buy any required reservations. For the French portion of the Luzern to Paris trip, these reservations are limited per train, so you would need to buy this when you bought the pass. I don't know if reservations are limited on the Eurostar. Reservations are not limited on Italian trains, so you can buy them close to travel.

Note that if you're just buying regular tickets, they will include any required reservations. This is true whether you buy them in advance or at the last minute; whether you buy them at a station or online; and whether the tickets are full price or discounted. Once you have a reservation, you can only board the train you have reserved. If you want to take another train, even with a fully refundable and changeable ticket, you must change the reservation BEFORE boarding, or you'll get a hefty fine. All of this means that rail pass "flexibility" in countries like Italy and France is not what it used to be (Switzerland and Germany, where most trains are unreserved, are different).

EDIT: cross posting with the others - you see we're saying the same things!

Thank you, all of you! Currently, I'm in planning phase of next summer's trip :) so all of this information is great. One question - If I buy a pass, I'm paying only the reservation fee per train right? say 10 euros per person for now. Is that right? so for my itinerary :

Rome - Venice - 10 euros
Venice - Florence - Pisa - 20 euros
Pisa - Lucerne - 20 euros ( 2 or more changes probably)
Lucerne - Paris - 20 euros ( 2 or more changes probably)
Paris - London (Eurostar) - 10 euros

This way, I know what is the exact cost of travel with pass vs booking individual tickets. Am I missing any other fees/charges that you have to pay apart from reservation charges + cost of Eurail pass?

Also, how early the tickets go on sale if I book them individually? That way, I can time it and see what comes cheaper.

When you are deciding between a Eurail Pass or point-to-point tickets, do not compare their respective costs on the RailEurope website. Check the cost of point-to-point tickets on trainline or loco2.com. You will probably come out ahead if you buy point to point tickets.

You don't have to wait until tickets for your dates go on sale to figure out what they will cost. You can check prices today for tickets as far in the future as each individual country allows (could be 90 days, 120 days, maybe more). That price should be pretty close to what you'll pay when your tickets actially go on sale, though there might be a modest fare increase for 2019. Don't forget that the individual fares you find online include the reservation fee when a seat reservation is required. I'm not certain, but I think the reservation fees in your earlier messsge are too low in some cases.

I believe the fares you see now (for travel in November-Janiary) will be so much lower than the cost of a railpass that there will be no doubt about your decision.

Rome to Venice, 10 EUR
Venice to Florence, 10 EUR
Florence to Pisa to Florence, no reservation fees if you use regional trains.
Florence to Milan, 10 EUR
Milan to Luzern, 11 EUR
Luzern to Basel, no reservation fee.
Basel to Paris, 25 EUR
Paris to London, 30 EUR.

These are all 2nd class reservation fees.