I tried to make some sentences which useusing new words I have learned from a book today.

I've always enjoyed learning the meanings of new words and their history behind itsthem.

I changed the tense to present perfect tense since you started the action "learning" in the past and it's continuing to the present.
I also changed "its" to "them" since "words" is plural.

And it seems to me that knowing how to use them precisely is alsois important, so here are the sentences I made and I hope these works can help me make an improvement.

I don't think "works" is a suitable word to use here, but I don't know what to substitute it with

Taking his poor economic situation into account, it is reasonable whythat he decided to work after graduating from school.

Personally I would use "it makes sense that he ..."

2. credence ¶
Her story about aliens is lacks credence for most people.

Or "For most people, her story about aliens is lacking credence"

3. gravitas ¶
He never fails to carry himself with gravitas whenever he attends activities in public.

"When" is correct, but you can substitute it with "whenever" to convey the meaning that he does something (carries himself with gravitas) every time something happens (he attends activities in public).

4. gravitate toward ¶
During summer vacations, people's social life often gravitated toward water activities, where you can stay cool and have fun with your families and friends.

Or "During summer vacations, people's social life often center around water activities..."

6. advocate ¶
He is a well-known advocate of pacifism, who refuses to ever fight ever for any reason.

A lot of people surarounding me have been infected this month.

I think "around" is a better word since "around" means nearby or close

I always enjoy learning the meanings of new words and their history behind its.

You could also say, "... and the history of them."

And it seems to me that knowing how to use them precisely is also important, so here are the sentences I made and I hope theseis works can help me make an improvement.

"...can help me improve," would sound more natural, but what you wrote works.

Taking his poor economic situation into account, it is reasonable whythat he decided to work after graduating from school.

2. credence ¶
Her story about aliens is lacks credence for most people.

4. gravitate toward ¶
During summer vacations, people's social life often gravitated toward water activities, where you can stay cool and have fun with your families and friends.

Just changed the tense of the verb.

6. advocate ¶
He is a well-known advocate of pacifism, who refuses to ever fight ever for any reason.