Mostly interchangeable, slight register difference
我刚到 and 我刚刚到 both mean "I just arrived". In practice, 刚 sounds a fraction more neutral and matter-of-fact; 刚刚 adds slight emphasis on the recency ("just a moment ago, literally"). In writing, 刚 dominates. In casual speech, 刚刚 is common especially when the speaker wants to emphasize how recent the event was.
我刚到家,刚刚才打开电视。
Wǒ gāng dào jiā, gānggāng cái dǎkāi diànshì.
I just got home and just now turned on the TV.
刚才 is a different beast
刚才 (gāngcái) functions as a time word (noun-like) meaning "a moment ago". Unlike 刚 and 刚刚, 刚才 can stand alone: 刚才你说什么 (what did you say just now?). 刚才 commonly appears at the start of a sentence or after the subject, and it refers to a specific recent moment in the past. 刚 and 刚刚 are adverbs that must attach to a verb; 刚才 is a time noun that floats.
刚才我在厨房,没听到电话。
Gāngcái wǒ zài chúfáng, méi tīng dào diànhuà.
I was in the kitchen just now and did not hear the phone.
Position: 刚/刚刚 must follow the subject
刚 and 刚刚 go after the subject and before the verb: 他刚走 (he just left), 我刚刚吃完 (I just finished eating). Putting them before the subject is ungrammatical: 刚他走 is wrong. 刚才 has more flexibility and can appear at the start of the sentence: 刚才他走了 is fine.
刚 with duration expressions
刚 pairs with duration expressions to mean "only just" in the sense of recent: 我刚来三个月 (I have been here for only three months, recently arrived). 刚刚 also works: 我刚刚来了三个月. With these duration uses, 刚才 does not fit; 刚才三个月 is wrong because 刚才 refers to a specific recent moment, not a recent period.
我刚刚工作了半年,还在学习。
Wǒ gānggāng gōngzuò le bàn nián, hái zài xuéxí.
I have only been working for half a year and am still learning.
Negation and question forms
Negating 刚 or 刚刚 with 没 is tricky because "just did not X" sounds odd in Chinese. Typically you rephrase: 我刚没有吃 is unusual; 我还没吃 (I have not eaten yet) is more natural. For questions, the affirmative form is standard: 你刚吃了吗 (did you just eat?). Substitute 刚才 for the more natural: 你刚才吃了吗 (did you eat just now?).