Are each of the followings correct to use? How a native speaker would express the same idea?
a) You should be careful about the people you mix yourself with.
b) You should be careful about the people you surround yourself with.
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The simple usage answer is that you "mix with people" you don't "mix yourself with people". However you do "surround yourself with people". So the first sentence is incorrect. The meanings of "mix with" and "surround yourself with" are different: using "surround" suggests the person is acting as a leader in the group and not a peer.
The deeper answer is that giving advice depends greatly on the relationship between the two people. The "no magic words" principle applies: There isn't a set phrase that native speakers use. There would be a lot of linguistic and non-linguistic context.