After a first taste of fame in a successful local theatre production, she went on to become a well known actress. 

参考译文：在当地一部成功的戏剧作品中首次体验到一举成名滋味之后，她成为了一名著名的女演员。

a first taste of fame 是“第一次/首次体验一举成名的滋味/经历”

taste  n.滋味;经历，体验，尝试

But having had a taste of the big time, he won't want to go back to playing in the reserves. 

但是，有过一举成名的经历以后，他就不会再想回去当替补了。

This voyage was his first taste of freedom.  

这次航行使他初次体验到了自由的滋味。