"Am I able to do this without a car as I would prefer not to get one unless necessary."
"Any information on transport Paris-Ghent, Ghent-Antwerp and Antwerp-Amsterdam would be appreciated."
Doing it without a car is not only possible, it's the only sane way! You will take trains between these cities.
To find schedules from Ghent to Antwerp, look at the Belgian rail site: http://www.belgianrail.be/en/Default.aspx. These trains run frequently, are unreserved, are cheap, and neither need nor allow reservations.
To find trains from Paris to Ghent and from Antwerp to Amsterdam, look at the Belgian rail international site: https://www.b-europe.com/Travel.
From Antwerp to Amsterdam, there are two options. There are Intercity (IC) trains, some with changes and some without, that will be slower (1:52 or 2:21 depending on the particular run). And there are Thalys trains that will be faster (1:12). The catch is that Thalys trains booked ahead as non-refundable tickets are cheap (actually a bit cheaper than IC trains, as low as €29 second class); however, the Thalys price goes up, a lot, closer to travel. For this coming Monday February 6, some second class Thalys tickets are €73 and some are sold out. The IC trains are a fixed price whenever you buy them (€35.40 second class). So, if you're sure of your plans, you can book a Thalys; if not, you can wait and take an IC.
By the way, on these trains second class is fine - much more spacious than coach on an airplane.
I don't know all the particulars about Paris to Ghent, but you can look on the Belgian Rail International website and see your options. Look at prices for tomorrow and prices for a few months out, to see which trains change in price and/or may sell out, and which ones have fixed prices.
If you want to learn more about trains, The Man In Seat 61 is a great resource, but may be a bit overwhelming as a place to start. Here's his page on Belgian trains, to start you off: http://seat61.com/Belgium.htm
"Also thanks very much tonfromleiden for your information regarding the bilingual signs at the stations - it may have been very confusing had I not known."
Yes it is potentially confusing; look in your guidebook for lists of the names of cities in the two languages. You'll find the train announcements switch languages as they pass through the different regions. Only the Brussels area is bilingual; in other parts of the country, they will announce the stations in the language of the region you are in. So, in Flanders, my train's next stop was announced as Luik; when we arrived in Wallonia, they said we had arrived at Liege. You just have to know that that's the same city.