In Cantonese, you use 先 to mean "first." For example:
Eat first => 先食
In Mandarin, most people use 才 so it becomes:
But sometimes people also use 先 even with Mandarin, so what are the guidelines in Mandarin for using 才vs 先?
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才 and 先 are both very common words in Mandarin. Like you said, 先 means first in Cantonese, but it is the same in Mandarin.
先 means first or previous in time(or order), or means earlier than ..., or means in advance, or means before ..., the opposite word is 后(means later).
examples: 早先,先前 both mean previously(or in the past). 先人 means ancestor
她比我们先到了那里——She got there before us.
才 has two kinds of character.
When it's used as noun it means ability or talent or gift. (example:你真有才——you really have talent! 多才多艺——be gifted in many ways.)
When it's used as adv, it means just now (example:刚才——just now, 才开始——just start),
or means things happen or finish so late (example:你怎么才来——Why are you so late!, 他昨天才走——He doesn't leave until yesterday).
or means short in number (example:你才十八岁——you are only 18 years old!)
or means only ...when (example: Only when you try hard, you can get good results——努力才有好结果),
or mean emphasize (example: I'm not buying what you said at all——我才不信你说的呢)
this is all I know, hope it helps
Eat first => 先食
It is correct, because 先 means 'first' in both Cantonese and Mandarin, so 先食 in Cantonese means the same as 先吃 in Mandarin
But in Cantonese, '先'(shortened for 先至) is also the counterpart of '才' in Mandarin
 [粵] [adv] only then (shortened for 先至)
Only after you finish them all, then you can get off work
Only after I bought it, then I realized it was a fake
Your example sentence: "努力先有好结果" is a valid shortened version of "努力先至有好结果"
才vs 先: When to use each one in Mandarin?
Aside from using '先' as the shortened form of '先至' in Cantonese, the usage of 先 is basically the same in Mandarin and Cantonese.
They have different meanings, I'm not familar with Cantonese, but in Mandarin, "先" means first, before, "先吃" and "先食" have the same meaning of "eat first". "才" indicates doings something late than normal, "我三点才吃午饭" means I had lunch at three pm, which delivered a message that I was busy at lunch time, and have to eat lunch late.
才 could mean 'a little late' or 'just now' according to the context.
才吃 means you just finish eating and possibly implying it's late according to the speaker's tone. The intonation plays a key role here.
This is hard for a Chinese language learner to distinct whether the speaker has the implication of 'late'.
For example, 你才吃完啊？ could just simply mean you just finish eating. Or if you put an emphasis on 才 while you speak, it means you just finish eating but emphasize more on "you are actually late" for eating. It could be a complaining sometimes.
先 just means 'first', 'earlier'. It's not that hard as 才.
先吃 means you eat a bit earlier than others/normal; you eat first. Like in 我先吃；你先吃；