My understanding (and I've never used the CT-S1 only read up about it) is you'll be able to (fairly) easily route the keyboard signal via a Micro USB cable to the computer.  You will need to the correct drivers installed for the keyboard, and then PTQ (Pianoteq) or a DAW hosting PTQ will "see" your MIDI USB device and can be configured to route the keys played to PTQ.

It's also theoretically possible that the "Aux in" port allows your computer sound card's "Line out" port via an 1/8" TRS Audio Cable (marketed under various names like "Stereo audio cable" or "Stereo headphone cable" etc.) to return the PTQ sound to the keyboard (as the keyboard's marketing would imply is possible), but that same online literature doesn't make it clear if that will work automatically or only with special configuration settings in the drivers or through some other means.  You'll only know through testing: happily plugging a 1/8" Audio Cable should be generally safe for experimentation as that's usually quite a low voltage connection.  You might need to depend on your computer's speakers rather than the keyboard's speakers if that doesn't work.

There is a Bluetooth Antenna pack (that seems to be a separate purchase) which works instead of the USB connector: it's possible that offers different connectivity options for routing the sound back to the speakers; however, that's only a guess.  You might from the driver configuration for the USB connection be able to route audio directly to the speakers, if that's supported in the drivers, such behavior would be atypical compared to other keyboards on the market but not at all impossible/unprecedented.  If that's the case, you'll be able to select your keyboard speakers as the sound output device in PTQ (or your DAW) whenever the keyboard is plugged in.  Against, that's just a guess that it supports, I can find no evidence for or against that functionality online through quick reading of a small portion of the documentation and marketing literature.

One final observation: when using certain operating systems (like Windows 10), Bluetooth is notoriously unreliable in some situations.  This varies from computer to computer, but in general, USB will be inherently more reliable and require less tech debugging to setup and keep working.  While bluetooth may offer different and more attractive connectivity options, it could come with less reliability that might require frequently restarting the keyboard, the bluetooth connection, and PTQ.

It's also theoretically possible that the "Aux in" port allows your computer sound card's "Line out" port via an 1/8" TRS Audio Cable (marketed under various names like "Stereo audio cable" or "Stereo headphone cable" etc.) to return the PTQ sound to the keyboard (as the keyboard's marketing would imply is possible), but that same online literature doesn't make it clear if that will work automatically or only with special configuration settings in the drivers or through some other means.  You'll only know through testing: happily plugging a 1/8" Audio Cable should be generally safe for experimentation as that's usually quite a low voltage connection.  You might need to depend on your computer's speakers rather than the keyboard's speakers if that doesn't work.

There is a Bluetooth Antenna pack (that seems to be a separate purchase) which works instead of the USB connector: it's possible that offers different connectivity options for routing the sound back to the speakers; however, that's only a guess.  You might from the driver configuration for the USB connection be able to route audio directly to the speakers, if that's supported in the drivers, such behavior would be atypical compared to other keyboards on the market but not at all impossible/unprecedented.  If that's the case, you'll be able to select your keyboard speakers as the sound output device in PTQ (or your DAW) whenever the keyboard is plugged in.  Against, that's just a guess that it supports, I can find no evidence for or against that functionality online through quick reading of a small portion of the documentation and marketing literature.

One final observation: when using certain operating systems (like Windows 10), Bluetooth is notoriously unreliable in some situations.  This varies from computer to computer, but in general, USB will be inherently more reliable and require less tech debugging to setup and keep working.  While bluetooth may offer different and more attractive connectivity options, it could come with less reliability that might require frequently restarting the keyboard, the bluetooth connection, and PTQ.

which cables to connect to both laptop and Casio CT-S1 so
that I can play the Pianoteg sound thru the speakers on the Casio Keyboard

For this part of the question, people in the Casio forums might know more.

Based on looking at the manual and this post, my guess is that it won’t work in any straightforward way.

You’ll be able to play Pianoteq from the keyboard, but getting the sound to come back out of the CT-S1 speakers will be problematic. It looks to me like you can’t route the sound from a computer back to the CT-S1 over USB; you’d need to use an audio cable to the input jack (which might or might not work, I can’t tell). But then you would run into a problem with what’s called latency: delay between pressing a key and hearing the sound. That’s just an inherent problem with Windows machines that requires another piece of gear, called an audio interface, to work around.

But, there might be another way out. It looks like the CT-S1 has a bit more functionality, possibly including what you want to do, with an iPad or iPhone; and Pianoteq can run on those. If you have either (or wouldn’t mind getting one), I suggest going to those Casio forums for more guidance on whether you can do what you want.

hd graphics 4400    4gb Rm   500 GB HDD

My Keyboard is a new Casio CT-S1  with 76 keys

Can anyone advise please if this Laptop is OK  - If so Can you advise which cables to connect to both laptop and Casio CT-S1 so

that I can play the Pianoteg sound thru the speakers on the Casio Keyboard

Also which basic Pianoteq would you recommend

I would appreciate any advice which can be given      Thanks