Rashmi,
It's not necessary to buy advance tickets on Regionale trains. It's just as easy to buy them at the local stations and then validate (time & date stamp) the tickets prior to boarding the train on the day of travel. DON'T forget to validate or you may face hefty fines which will be collected on the spot!
There are numerous trains every day from Florence to Lucca, some direct and some with one change, with travel times as short as 1H:19M. You can see all the solutions on the Trenitalia website. That's a very easy trip! The station in Lucca is outside the walls, so you'll have a short walk (or Taxi ride), depending on where your apartment is located.
Just to clarify the situation in the Cinque Terre, this appears to be a case of someone irresponsibly reporting measures that were being discussed, but not yet decided. Those reading the article assumed these changes were a "done deal" and that's not the case. There are currently NO new "restrictions" in place in the C.T. and it's not likely there will be any significant new measures implemented, at least not this year. There are many different stakeholders, and these discussions will probably go on for quite some time as they will all have to agree on a solution.
The trip from Lucca to Venice is also very easy. Note that your destination station will be Venezia Santa Lucia. The quickest and easiest rail trips on that route appear to be via Firenze S.M. Novella, and then high speed to Venice. One point to note is that you can use either Trenitalia or Italo Treno high speed trains from Florence to Venice. The service is the same, so use whichever one has the best timing and prices. You can save money by purchasing advance tickets on the high speed trains, but keep in mind that the cheapest tickets are non-refundable and non-changeable once purchased and are specific to one train, date and departure time, so choose carefully before hitting the "Buy" button.