If someone puts my SIM card into their phone, will they see my text messages? If someone takes my SIM card and puts it in their phone, will they be able to read my text messages?

What will they be able to access in terms of my personal information other than make outgoing calls using my SIM card and receive my text messages?
Text messages are stored on your phone, not on your Sim. 

Therefore, if someone puts your Sim card into their phone, they will not see any text messages that you have received on your phone, unless you have manually moved your SMS's to your Sim.

As long as you haven't saved your contacts to your Sim card, they will not have any personal info at all.

Have in mind, that there are apps like WhatsApp, cashapp telegram, and many others that their only authentication system, is your phone number. Therefore if you had accounts with any of the above, chances are that anyone with your Sim card can now go ahead and set up a WhatsApp or other accounts with your number.

I therefore suggest that you contact your phone company and transfer/cancel your number as soon as possible.