What it means
ようと思う expresses a personal intention or plan to do something. It is formed with the volitional form of a verb, followed by と and 思う. The speaker presents the action as something they are considering or intending, so it is softer than a firm promise. The polite counterpart is ようと思います.
Usage note
Do not confuse ようと思う with つもりだ. ようと思う emphasizes a current thought or decision about an action, while つもりだ more directly describes an intention or plan.
How to use it
Verb (plain volitional form)と思います
私は来月京都へ旅行しようと思います。
I am thinking of traveling to Kyoto next month.
Verb (plain volitional form)と思う
私は来月京都へ旅行しようと思う。
I am thinking of traveling to Kyoto next month.