It could help to think of the chained verbs as a series of actions or if you are familiar with Japanese adverbs, think of the final verb in the compound as the actual action and the verbs preceding it in て-form as an adverbial describing how.
ルーカスさんが車でアパート探しに連れて行ってくれた。
Think of 「連れて行く」 as "to go (somewhere) leading (somebody)" (i.e. to take somebody along).
ルーカスさんが車でアパート探しに連れて行ってくれた。
Now think of the previous verb as an adverbial describing how 「くれる」. "To do (something) for (somebody) going (somewhere) leading (somebody)". Somebody can refer to oneself.
Mr. Lucas, with a car, to apartment hunting, did the favor of taking himself along.
Or in more natural English:
Mr. Lucas joined me in my hunt for an apartment with his car.
Your sentence:
ルーカスさんが車でアパート探しに連れてた。
It is the colloquial form of 連れていた and has a completely different meaning from 連れて行ってくれた. I'll assume you meant 連れた.
Mr. Lucas, with a car, to apartment hunting, led (something or somebody).
This is much more ambiguous from when 行く and くれる are used. First of all, it doesn't mean taking somebody along but only leading something. The omission of くれる also leaves it unclear what was done to what.
連れてた
-> You meant to say 連れて行った, perhaps?