How can I highlight one word in a sentence in order to copy and paste it?

Control click or double click on a word should select it.

Triple click should select the entire paragraph.

If that isn't working, is it in all applications (which apps have you tried it in)?

Are you using the built-in trackpad or a mouse or other device?

Give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

I just tested it in Notes on my MacBook Pro and couldn't find any issue. It worked with both clicking and sliding and with clicking and then moving the cursor and shift-clicking.

You may want to contact Apple Support to see if they have any ideas. There is a Get Support link at the top right of this page.