Which one is correct and more common? is there better way?
- Let’s start speaking English.
- Let’s start to speak English.
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Regarding the use of the gerund or to-infinitive after the verb to start, it seems to be a matter of one's personal preference, at least according to the BBC Learn English site.
At the same time, on some English language forums, you may read that start to do can apply to a regular activity and the gerund to sudden bursts. I could not find any reliable reference supporting that, though.
As for the use of the gerund vs. to-infinitive after "let's start", I think that a particular context of suggesting something being set in motion might affect your choice.
For example, the verb to speak may mean to discuss/convey ideas, in which case the use of the to-infinitive would be quite appropriate:
Let's start to speak business, politics, etc.
With this in mind, the phrase "Let's start to speak English" might be taken sooner as an invite to a discussion about the English language, than as a suggestion of just speaking it, where the gerund should be used:
Let's start speaking English
For examples of the to-infinitive usage after "Let's start", see here.