Od ta chvíle nacházíme jeho stopy v nesčetných zemích skoro všechny světové kontinenty.

The main verb is "nacházíme" 1pl. present tense form of nacházet "to find (regularly)"

The subject is the implicit "my."

The verb nacházet requires a direct object; "jeho stopy"(his traces) qualifies for this; všechny světové kontinenty would work grammatically as a direct object, but it would not make less sense (we regularly find all the world continents).

Od "from" takes the genitive case; with "ta chvíle" the prepositional phrase reports time 'from this moment.'

V nesčetných zemích 'in numerous countries' is a locative prepostional phrase reporting the location where we find his traces. Všechny světové kontinenty best fits this whole sentence if it is considered as adding information to "v nesčetných zemích," thereby reporting "in numerous countries OF almost all the world continents." This can be done by the genitive case.