The nouns きっかけ and 契機 both indicate taking advantage of a situation in a positive light to bring about what follows.
きっかけ can literally be translated as "chance/cue/motive/occasion" or more loosely translated as "trigger/prompt."
Orthography Note: きっかけ is only seldom spelled in Kanji as 切っ掛け.
Synonym Note: Although 機会 and チャンス both translate as "chance," they are one-time opportunities that you don't want to miss out on, whereas きっかけ is the "chance" to start something that is long lasting.
What was it that triggered you to start studying Japanese?
The motive to improve myself, the course of doing so, and the results
The motive and process for how one becomes able to speak English differs from person to person, but there are several points of commonality.
In the grammar pattern ～をきっかけ【に（して）・として】, it is translated as "taking advantage of.../inspired by..." What follows is something that the speaker/agent is now actively doing.
Prompted by this case, major revision regarding the weapons that regular officers possess became pressing.
With the accident as a start, I realized the love my family had for me, and my solidarity with them strengthened.
I took advantage of getting married to change jobs.
Young people are being involved in crimes with social media as their motive.
You may also see the pattern ～がきっかけで, which indicates that something happens due to a trigger. It is most appropriate in contexts regarding happenstance.
That single comment was what prompted his sense of reasoning to completely collapse.
I became interested in Japanese with Pokemon as the motive.
The noun 契機 meaning "opportunity" is a rather formal word that is usually only seen in the written language. Similarly to きっかけ, it can be seen in the pattern ～を契機【に（して）・として】.
Particle Note: The use of として over に（して） is both a matter of formality and semantics. While the former is more formal by default, it also implies that the action bringing brought about is the final solution that the speaker/agent is working towards. On the other hand, に（して）will imply that the action bringing brought about is very much on going and still might have the possibilty of changing given the right (change in) motive.
Violence is occurring over the suppression of speech.
Taking the (act of) terrorism as a motivation, security has been greatly enhanced.