You are too early for reservations for that date, they are currently only available up to the end of March, check back in a few weeks. Trains do not generally sell out far in advance, many will still have availability on the day of travel.

There are no direct trains Paris - Lucerne, you will have to change at least once.

Only the TGV from Paris needs reserving.

You have 2 options;

The cheapest is to reserve Paris - Strasbourg or Paris - Mulhouse Ville, this will cost €10 or €20 per person (there are limited €10 reservations, once gone it is €20 until the train is full)

From that you connect to an unreservable train to Basel SBB and Swiss train to Lucerne.

More expensive is Paris to Basel SBB, reservation is €39 per person, change there for Swiss train to Lucerne.

Reservations on some Swiss trains are possible but they are not necessary and most passengers do not make them.

Best place for French reservations is now https://www.raileurope.com/en they do not charge a booking fee.

Thank you so much for your kind response. I am so grateful !!! I already knew about the cheaper option route and had it saved in my trips. But thanks so much for confirming I am too early yet. I have pinned your link  

by any chance do you know where I book my seat reservations for Lucerne to Venice ? 
 

thanks again !! 

The same site also provides Swiss-Italy and Italian reservations.

See this comprehensive guide to pass reservations for more details.

https://www.seat61.com/interrail-and-eurail-reservations.htm

Thanks so much for that !  I have read that article a lot.  It's a bunch to get my head around.  Thanks again !

If you are travelling first class it is always worth checking the OBB site/app. If you add an Interrail/Eurail pass to each passenger as a discount card and check for tickets (not seat reservations) it will show trains and, where available, give free reservations ( small processing fee may apply). They cover a range of countries. Lounges in Austria ( and UK) are available to pass first class users.