Can someone access my old texts or voicemails if I give them my SIM card?

The biggest risk is account recovery. Many services send login or password-reset codes by SMS, so whoever controls your old number could potentially reset accounts linked to it. Change the phone number on important accounts, enable an authenticator app where possible, remove the number from your Apple account, and ask your carrier to transfer or close the line properly.

Usually, no. Modern phones generally store messages on the device or in the associated cloud account rather than on the SIM card. They should not be able to read your old iMessages or previously stored texts just by inserting the SIM. Some SIM cards can store a small number of basic SMS messages, but that is uncommon with current smartphones.

They will receive new calls and text messages sent to that phone number after the SIM is activated in their phone. They will not automatically receive your old iMessages because those are connected to your Apple account and devices, but make sure the number is removed from your Apple account and iMessage settings first.

Voicemails are tied to your phone number and carrier account, not normally stored on the SIM. However, the other person may be able to access your voicemail if they know or can reset the voicemail PIN. Before handing over the number, delete saved voicemails and change the voicemail password or PIN.