You don't need to study axioms in order to accept them, but once you accept them, then you must accept any soundly derived conclusion from them. Belief doesn't need to be logically consistent, but knowledge does.

As for investing significant time and energy, I would say that that depends on things such as the length of the chain of reasoning or the difficulty/cost of testing a hypothesis or how closely observations match your intuition. Some knowledge is cheap to acquire, such as "the sun rises in the east", because we can observe it directly and we can clearly identify the direction of east and the sun's path in the sky is very stable from day to day.

Id say thats quite obviously the way it works. How would you have knowledge on something if you havent researched it thoroughly? If you are just parroting what someone else told you its no better than hear-say.