Kad budem našla rešenje, poslaću ti poruku.

Because this sentence has two future actions, and the first one must happen before the second one:

• kad budem našla rešenje = when I find / when I have found the solution
• poslaću ti poruku = I will send you a message

In Serbian, when a future action in a subordinate clause is completed before another future action, Serbian often uses Future II in the subordinate clause. That is what budem našla is.

So the structure is:

• Kad budem našla rešenje, ... = When I have found the solution, ...
• ... poslaću ti poruku. = ... I will send you a message.

In natural English, we usually just say When I find the solution, I'll send you a message.

Budem is the 1st person singular form used in Future II.

The Future II is formed with:

• budem / budeš / bude / budemo / budete / budu
• plus the past participle of the main verb

So here:

• budem našla

This is different from the ordinary present-tense auxiliary sam. You would not say Kad sam našla rešenje if you mean the future.

A useful mini-paradigm:

• budem našao / našla
• budeš našao / našla
• bude našao / našla

Because the speaker is female.

In Serbian, the past participle agrees with the subject in gender and number.

So:

• female speaker: našla
• male speaker: našao

If a man were saying the same sentence, it would be:

• Kad budem našao rešenje, poslaću ti poruku.

Poslaću is a normal contracted form of the future tense:

• poslati + ću -> poslaću

This is very common in Serbian, especially when the verb comes at the beginning of the clause.

You can also express the same future with the separate auxiliary, but then the word order changes:

• Ja ću ti poslati poruku.

What you generally cannot do is start the clause with bare ću:

• Ću poslati ti poruku — not natural/standard

So in this sentence, poslaću ti poruku is the most natural choice.

Ti is the dative form of you, and here it means to you.

So:

• poslaću ti poruku = I will send you a message
• more literally: I will send a message to you

This is very common in Serbian with verbs like:

• dati = to give
• reći = to tell
• poslati = to send
• pokazati = to show

Examples:

• Daću ti knjigu. = I’ll give you the book.
• Reći ću ti sutra. = I’ll tell you tomorrow.

Rešenje is the direct object of našla, so it is in the accusative singular.

However, rešenje is a neuter noun, and for many neuter nouns, the nominative and accusative singular are the same.

So:

• nominative: rešenje
• accusative: rešenje

That is why the form does not change.

Also, Serbian has no articles, so there is no separate word for the or a. Whether it means a solution or the solution depends on context.

Because poruku is the accusative singular form of poruka.

• poruka = nominative singular
• poruku = accusative singular

Since poruka is the direct object of poslaću, it must be in the accusative:

• poslaću poruku = I will send a message

This is a very common pattern with feminine nouns ending in -a:

• knjiga -> knjigu
• poruka -> poruku
• kafa -> kafu

Here kad means when.

• Kad budem našla rešenje, poslaću ti poruku. = When I find the solution, I’ll send you a message.

This is different from ako, which means if:

• Ako budem našla rešenje, poslaću ti poruku. = If I find the solution, I’ll send you a message.

The difference is:

• kad suggests the speaker sees it as something that will happen, sooner or later
• ako presents it more as a condition or possibility

Yes — that is also very common and natural.

Many speakers use:

• Kad nađem rešenje, poslaću ti poruku.

This uses the perfective present (nađem) with future meaning after kad.

Compared with kad budem našla rešenje, the version with budem našla can sound a bit more explicit about the action being completed before the next one. The version with nađem is often simpler and very common in everyday speech.

So both are useful to know:

• Kad nađem rešenje...
• Kad budem našla rešenje...

Because the sentence is about completed, one-time actions:

• naći = to find, successfully reach the result
• poslati = to send, complete the sending

That fits the meaning perfectly:

1. first, the speaker finds the solution
2. then, the speaker sends a message

If you used the imperfective partners instead, the meaning would change:

• nalaziti = to be finding / to find repeatedly
• slati = to be sending / to send repeatedly

In this sentence, the focus is on the result of each action, so perfective verbs are the natural choice.

Yes, the word order can change.

When the kad-clause comes first, Serbian normally uses a comma:

• Kad budem našla rešenje, poslaću ti poruku.

You can also put the main clause first:

• Poslaću ti poruku kad budem našla rešenje.

Both mean the same thing.

Serbian word order is more flexible than English, but clitics such as ti have preferred positions, so not every rearrangement sounds natural. That is why poslaću ti poruku sounds good, while random reshuffling may not.