Learn Polish: Mój syn lubi ten jogurt. - My son likes this yoghurt.

The verb lubić (to like) takes the accusative case for its direct object. However, jogurt is a masculine inanimate noun. In Polish, masculine inanimate nouns and their pointing words like ten have the exact same form in the accusative case as they do in the nominative case.

Could I use the verb podoba się here instead of lubi to mean 'likes'?

You could, but the sentence structure would completely change. Podobać się means 'to appeal to' and requires the person who likes the thing to be in the dative case. You would have to say Mojemu synowi podoba się ten jogurt (This yoghurt appeals to my son). Using lubić is often easier here because it matches the English subject-verb-object order.

If my son liked a feminine noun, like zupa (soup), how would the end of the sentence change?

Because the verb lubić forces the object into the accusative case, a feminine noun and its pointing word would both change their endings. You would say tę zupę instead of ten jogurt.

Why is it mój syn and not mojego syna?

Mój syn is the subject of the sentence (the one doing the liking), so it stays in the basic dictionary form, which is the nominative case. You only change it to forms like mojego syna when the son is the direct object of an action, for example if you said 'I see my son' (Widzę mojego syna).