Learn Hardcore Japanese: tenki ga hakkiri sinai node, kasa wo motte ikimasu. - Since the weather is uncertain, I’ll bring an umbrella.

が marks 天気 as the subject of the verb phrase はっきりしない and simply states a fact: “The weather isn’t clear.” If you used は (天気ははっきりしない), you’d be making 天気 the topic and possibly implying contrast (“As for the weather, it isn’t clear, but…”). Here, が keeps the focus on the weather’s condition itself.

What does はっきりしない mean, and why use しない instead of です?

はっきり is an adverbial noun meaning “clear(ly)/distinct(ly).”
Combined with the verb する, it becomes はっきりする (“to become clear”).
しない is the plain negative form of する, so はっきりしない means “does not become clear” or “is not clear/decided.”
You could also say はっきりしていない for a more continuous feel (“isn’t staying clear”).

Why is ので used here instead of から to express “because”?

Both ので and から give a reason, but:

ので is slightly softer and more explanatory. It often appears in both polite and plain contexts.
から is more direct and casual.
In our sentence, ので links the unclear weather to the decision to bring an umbrella in a gentle way:
“Since the weather is uncertain, I’ll take an umbrella.”
You could replace it with から without breaking the grammar: 天気がはっきりしないから、傘を持って行きます。

Why is the verb phrase 持って行きます used instead of simply 持ちます?

持って is the て-form of 持つ (“to carry”). When you attach 行く (“to go”) after it, you get 持って行く, meaning “to go carrying (something)” or “to take (something) with you.” In polite present tense, it becomes 持って行きます.

傘を持ちます just means “I will carry an umbrella.”
傘を持って行きます means “I will go, carrying an umbrella” (i.e. take one along to where I’m going).

Could I say 傘を持って来ます instead of 持って行きます?

Use 持って来ます when someone is bringing something toward the speaker’s current location. Use 持って行きます when someone (in this case, the speaker) is going elsewhere and taking something with them.

If you’re at home and you ask a friend, “Can you bring me an umbrella?” you’d say 傘を持って来てください.
If you’re leaving your house and telling someone else, “I’ll take an umbrella (with me),” you say 傘を持って行きます.

Is the sentence fully polite, casual, or mixed?

It’s a bit mixed:

しない is the plain negative form.
行きます is the polite positive form.
Mixing plain negatives with polite affirmatives like this is common in everyday conversation.
For a fully casual version, say:
天気がはっきりしないので、傘を持って行く。
For a fully polite version, say:
天気がはっきりしませんので、傘を持って行きます。

What other ways can I express “the weather is uncertain”?

Here are some alternatives:

天気が不安定だ (“The weather is unstable.”)
天気が読めない (“I can’t predict the weather.”)
雨が降るかどうか分からない (“I don’t know if it will rain.”)

Why is there a comma after ので in Japanese?

Japanese often uses commas (、) to separate clauses for clarity. After a conjunction like ので, the comma signals the end of the “reason” clause and the start of the “result” clause, mirroring how you might pause in speech:
[Reason]、[Result].