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Depending on the brand of device, you should be able to long press a thread of texts (to and from one contact) and a trash bin will appear while the thread is highlighted. Touch the bin and confirm.
For a single text on my Motorola, long pressing brings up a selection box including Delete.
You can delete everything by opening the Message app, find the menu button, usually a three dot symbol, and Delete All.
Let us know if your device is different.