Кое-кто считает, что такое опоздание не страшно, но в следующий раз лучше предупредить заранее.

Кое-кто means someone / certain people / some people. It often suggests that the speaker has a person or group in mind but does not name them.

Compared with кто-то:

• кто-то = a more neutral someone
• кое-кто = someone in particular, but I’m not saying who

In this sentence, Кое-кто считает... is like Some people think... or There are people who think...

Because кое-кто is grammatically singular in Russian, even if in English we may translate it more naturally as some people.

So Russian uses a singular verb:

• кое-кто считает
• кто-то думает
• никто не знает

This is normal with these pronouns.

Yes, считать has more than one meaning.

Common meanings are:

• to count
• to consider / to think / to regard

In this sentence, it means to think / to consider:

• Кое-кто считает, что... = Some people think that...

This is a very common use.

Here что means that and introduces a subordinate clause:

• считает, что такое опоздание не страшно
• thinks that this kind of lateness is not serious

After verbs like думать, считать, знать, говорить, Russian often uses что to introduce what someone thinks, knows, says, etc.

The comma before что is required.

Такое means such a or this kind of.

It agrees with опоздание, which is:

• neuter
• singular
• nominative

So:

• такой = masculine
• такая = feminine
• такое = neuter
• такие = plural

Here такое опоздание means such lateness, this kind of delay, or being this late.

Because страшно here is not an adjective modifying опоздание. It is a predicative word meaning something like:

• not serious
• not terrible
• not a big deal

So:

• такое опоздание не страшно = such lateness isn’t a big problem

If you said страшное опоздание, that would mean a terrible delay / a terrible case of being late, where страшное is a regular adjective describing the noun.

So the difference is:

• страшное опоздание = a terrible lateness
• опоздание не страшно = the lateness is not a big deal

Because опоздание is the thing being evaluated in the clause:

• такое опоздание не страшно

It functions like the subject of the idea is not serious. Russian often uses the nominative noun with words like:

• важно
• трудно
• страшно
• понятно

So the structure is essentially:

• [noun in nominative] + [predicative word]

Here: опоздание stays nominative.

Опоздание comes from опоздать (to be late). It can mean:

• lateness
• being late
• a delay caused by arriving late

In this sentence, the best natural sense is being late / this lateness.

So такое опоздание is basically being this late or this kind of lateness.

В следующий раз is a fixed and very common expression meaning next time.

Here:

• в
  • accusative can be used for time expressions
• следующий раз is accusative singular

Because раз is masculine and inanimate, its accusative form looks like the nominative:

• nominative: следующий раз
• accusative: следующий раз

So literally it is something like on the next occasion, but idiomatically just next time.

This is a common impersonal way to give advice.

• лучше + infinitive = it’s better to... / you’d better...

So:

• в следующий раз лучше предупредить заранее
• next time, it would be better to let someone know in advance
• next time, you’d better warn them beforehand

The subject is understood from context. Russian often leaves it unstated when it is obvious.

Because the sentence is talking about a single complete action in the future:

• next time, do one thing: let someone know

Perfective verbs are very common when Russian refers to a one-time completed action.

So:

• предупредить = to warn / inform once, successfully
• предупреждать = to be warning / to warn repeatedly / habitually

Here the advice is about one future occasion, so предупредить fits best.

In many contexts предупредить does mean to warn, but here it is closer to:

• to let someone know
• to inform
• to give advance notice

For example, if you are going to be late, предупредить means telling the relevant person ahead of time.

So in this sentence it is not dramatic warning; it is more like notify someone in advance.

Заранее means in advance / beforehand and makes the idea explicit.

Even though предупредить can already suggest giving notice, adding заранее strengthens it:

• not just tell someone
• but tell them ahead of time

This combination is very natural in Russian:

• предупредить заранее

So the sentence strongly suggests: next time, let someone know before it happens or before you are too late.

Russian often omits the object when it is obvious from context.

So предупредить here really means something like:

• предупредить начальника
• предупредить преподавателя
• предупредить нас
• предупредить их

But the speaker does not need to say exactly whom, because the situation already makes it clear.

English does this too sometimes:

• Next time, let someone know in advance.

It is closer to next time, you should warn / you’d better warn / it would be better to let someone know.

Because of в следующий раз (next time), the advice is future-oriented. It is not mainly blaming the past action; it is giving a recommendation for the future.

So a natural interpretation is:

• Some people think this lateness isn’t a big deal, but next time it’s better to let someone know beforehand.