Seat61 describes the options very well. It's described as a journey from London, but you can just ignore the first part.

It also has some great advice on where and how to buy tickets.

https://www.seat61.com/Spain.htm#london-to-barcelona-by-train

Option 1 on this page, by day, is the only direct way to Barcelona.

Options 3&4 are the two night train options, which take you nearly to the border. Followed by local trains to Barcelona.

Of course there are countless other options if you want to stop off somewhere like Lyon, Nîmes, or Montpellier, which are all on the high speed route and worth a visit.

The Night train could be kind of fun for the kids, and you could get a 6-person compartment. It would be cheaper price for reservations than the TGV, and save you a night's accommodation, but it takes much longer and needs a change of train.

Reservations costs, (if you're using a rail pass) and how to book them: https://www.seat61.com/interrail-and-eurail-reservations.htm#France

And, just to be clear, there are three trains per day on the direct Paris-Barcelona route (Option 1) in summer. Seat61 only mentions the afternoon departure because you can connect from London. 

But there are also two morning departures.

It's too early to book for July, but you can have a look at prices up to March.

Just go on raileurope.com and search for Paris-Barcelona (or to Latour de carol or Cerbere if you want the sleeper).