3 month old with cough

Neena

Just want opinions if I should be rushing by baby to the doctor or not. Yesterday he started having a small cough and snotty nose. It sounded just like a normal cough that you would do if your throat had a tickle. And it wasn't constantly, just like ever little while. He did sleep more yesterday than usual and was still eating and making wet diapers normally. No fever and demeanor was normal. Still very smiley and not fussy. Today, seems the same as yesterday as far as eating and demenor and no fever. Big difference really today is his cough sounds a little worse. It's still not constantly but when he does cough it seems a little more gravelly. He has a little bit of wheeze in his breathing but it's not intense and doesn't seem like he's struggling to breath. It could very well just be a cold but this is my first baby and first time he's been sick. So I don't really know if it's worth trying to run to the doctor right away or if I should just keep a eye out for a few days and just make sure he's doing okay. Helpppp