I am writing a homework assignment and I am trying to write "I had to start bringing my umbrella".
I know that to say "I started eating" it would be 食べはじめる，or to say "I started drinking" it would be 飲み始める。
I have no idea how to start writing "I had to start bringing an umbrella".
This is what I think it is: 傘を連れて行き初めていけない
This would be in the context of: It was raining so much, I had to start bringing an umbrella because I did not want to get wet.
If you had to construct the sentence using 〜はじめる, that would be:
傘を持っていきはじめなければいけなくなった。
I had to start bringing an umbrella.
Since that is quite a mouthful (and not very natural despite its grammatical correctness), some other options:
In general, I agree with @mirka.
I would also say:
傘を持参するようになった
Firstly, note that 連{つ}れて行{い}く is used for living things not inanimate objects. Inanimate objects would use 持{も}って行{い}く. Also note that ～ていけない in this instance is missing either the initial いけない （持{も}っていかなければいけない), however this implies that it is an absolute must that you bring the umbrella with you (not so much being required for rain as you will get in trouble for not having it).
Something simple like 雨が降っていたから、傘を持っていきました (I brought an umbrella because it was raining) should work fine.
For completeness, I'll mention one other way to say "to start ~".
You can use verb (pre-masu) + 出す.
For example:
彼は走り出した
He started to run.
雨が急に降り出した。
It suddenly started raining.
However, I think the usage of this is much more limited than 〜はじめる. See this post where a Japanese person talks about a special nuance of "食べ出す"
Also, since this usage focuses more on the act of starting something at that moment, I don't think you could use it for your umbrella example. The "start" in "start bringing my umbrella" is more about a long term trend.
