보다
bo-da
보다 is about where you park your eyes
The picture A gaze stuck fast to one spot. Stuck to a screen it is watching; stuck to a bag it is guarding. 보다 is you bringing your gaze to something; 보이다 is the thing arriving in your eyes on its own.
Where the picture shows up in 보다
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eyes on the screen
어제 그 영화 봤어요?
eo-je geu yeong-hwa bwa-sseo-yo
Did you see that movie yesterday?
In a dark cinema your two eyes stay glued to the screen for two hours without coming loose
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eyes stuck to the bag
잠깐 가방 좀 봐 주세요.
jam-kkan ga-bang jom bwa ju-se-yo
Could you keep an eye on my bag for a second?
The owner has stepped away, but your gaze stays stuck to that bag the whole time
봐 주세요 can mean 'look at it' or 'watch over it'. With a bag it means guard it.
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try it and see
이거 한번 먹어 보세요.
i-geo han-beon meo-geo bo-se-yo
Go on, have a taste.
You take one spoonful into your mouth and raise your eyebrows, checking how it tastes
-아/어 보다 means giving something a try to see how it turns out.
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how it looks to you
제가 보기에는 좀 비싼 것 같아요.
je-ga bo-gi-e-neun jom bi-ssan geot ga-ta-yo
To me it seems a bit expensive.
The line of sight leaving your eyes ends on a price tag whose number looks unusually large
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you go through it
내일 면접 보러 가요.
nae-il myeon-jeop bo-reo ga-yo
I have an interview tomorrow.
You open the interview-room door, sit down facing the panel, and go through it yourself
Exams and interviews take 보다, never 하다: 시험을 봐요.
All of it in one picture
보다 splits on one thing only: where you park your eyes. Park them on a screen and you are watching a film; park them on a bag and you are guarding it; park them on a mouthful and you are trying it to see. Park them on a price tag and it becomes how the thing looks to you; park them on an interview and it becomes going through it in person. So with 보다, start by picturing a gaze that has come to rest somewhere.