"The good children like this animal."
Translation:Dobre dzieci lubią to zwierzę.
"dobrze" is an adverb (well)
"dobre" is the adjective - either neuter singular, or 'not masculine-personal plural' (what we have here).
"stary" is a masculine singular adjective.
"starzy" is a 'masculine personal plural' adjective (at least one man among... whatever the noun/pronoun was).
Look at the Nominative singular form.
If it ends with a consonant, it's probably masculine.
If it ends with -a, it's probably feminine.
If it ends with -ę, -o, -um, it's neuter.
And then learn the exceptions ;) Still, the above will be right in 99% of the cases. But some common words (mysz - feminine, tata - masculine, mężczyzna - masculine) will be exceptions.