ようと思う / おうと思う【ようとおもう / おうとおもう】

ようと思う says what you intend to do: "I think I'll", "I'm going to". The verb goes in the volitional form – たべよう食べよう, いこう行こう, しよう – followed by とおもう思う, or とおもいます思います in polite speech.

ようと思う or と思う

With the dictionary form, と思う is a guess about what will happen: "I think it will". Only the volitional makes it your own plan.

• × 来年日本に行くと思う (for "I'm thinking of going") → 来年日本に行こうと思う

ようと思う or ようと思っている

ようと思う is a decision you are making as you speak. ようと思っている is a plan you have had for a while.

Someone else's plan always takes と思っている, usually with そうだ or らしい, because you can't speak for their decision in the moment.

• × 彼女は会社を辞めようと思う → 彼女は会社を辞めようと思っている

A plan you dropped

In the past, ようと思った is something you meant to do, often followed by what stopped you.