I read recently in a piece of poetry the sentence ”从明天起， 我会做一个幸福的人“ － what is the difference between this and, say, "从明天开始？“. Is 起 significantly more formal? Could I use it in day to day writing? (Say a diary?)
To answer directly towards your questions,
- Is 起 significantly more formal?
No, it is not. It is commonly used in spoken language, casual writings and formal writings.
- Could I use it in day to day writing? (Say a diary?)
Yes, and it sounds perfectly natural.
- ”从明天起， 我会做一个幸福的人“ － what is the difference between this and say "从明天开始？“.
As the other answer suggested, 从明天起 means 'from ... onward; ever since ...' which is making a statement, a bold prediction or an oath for a future result. E.g.
从明天起我就再也不抽烟了。After today I will never smoke anymore.
从明天开始 means 'starting from tomorrow', which is most commonly used when you're telling an execution plan. E.g.
从明天开始我每天少抽一根烟，直到戒掉为止。Starting tomorrow, I'll be using one less cigarette every day until I completely quit.
In practice the difference is very subtle and people use them interchangeably. In situations where the difference really matters, people tend to use more elaborated and explicit way to disambiguate them.
- 从明天起 = "from tomorrow onward". Its tone is similar to "It's believed that from this day onward the US economy will enjoy a mild boost." Or "from tomorrow onward I will be an adult."
- 从明天开始 = "starting from tomorrow". Its tone is similar to "Starting from tomorrow I will sleep early and do my laundry."
I hope that makes it clear.
Yes, you can.
Using 起 in this poetry of Haizi is more rhythmical in my opinion.
For everyday spoken Chinese using "起" is a bit spurious, and using "開始" is natural.
For written or formal Chinese using "起" is quite okay. Contrarily, using "開始" here gets a bit sloppy.
起is more used in writing than speaking, though we still use that in normal talking. A really formal way, as I could immediately remember, is "自……始" or "自……启".