Is this the correct Chinese spelling? 走向更大的事物?

No, the sentence “走向更大的事物” make no sense.
走向 : walking towards
更大: bigger
的: normally used before noun.
事物: means “something”, doesn’t have a meaning of “aim” or “purpose”, which I guess you’re trying to say.
So, if you’re trying to say “walking towards more ambitious aims”,
then, the correct Chinese sentence is as follows:
走向更遠大(的)目標:
走向 : walking towards
更遠大(更大 is acceptable, but 遠大is better): more ambitious
(的)目標: aims. Here, with 的 or without 的，both are acceptable.
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The meaning of 事物 is all the physical things and all the matters, phenomena in a situation.
The meaning of 事務 is some chores, preparations, errands, etc. that someone must get it done.
Basic reason is it isn’t proper to use 更大 as an adjective of 事物.
compared with 事，事務/事物is rather formal. So you could say “I have some small 事 to deal with”, but you normally wouldn’t say“I have small 事務/事物 to deal with”.
And you can say this 事務 is challenging, this 事物 is very complicated, etc.
If you want to say “towards greater things”, it seems you have your aims in career. In this scenario, you are supposed to say 走向(有)遠(很/更)大的目標.

Go to the small fish?

The correct Chinese spelling of what? May I ask.