Q. Is a comma used after “But” or “And” at the beginning of a sentence?
A. Not usually. For example, both of the following would be correct:
But I don’t want to meddle.
And to avoid getting my feet wet, I wore boots.
The rule is the same as when the conjunction introduces an independent clause within a sentence (usually following a comma; see CMOS 6.22):
I’d tell you how to punctuate this sentence, but I don’t want to meddle.
I carried an umbrella, and to avoid getting my feet wet, I wore boots.
but
I carried an umbrella and, to avoid getting my feet wet, wore boots.
It wouldn’t be strictly wrong to add a comma after the conjunction in the second and fourth examples (see CMOS 6.26), but Chicago usually prefers to omit it unless the conjunction joins a compound predicate (as in the fifth example; see CMOS 6.32). The same logic could be extended to similar constructions that might otherwise call for two commas. For example,
But Jerome, we haven’t yet discussed commas.
Some editors would add a comma before “Jerome” (see CMOS 6.53), but unless you’re applying or enforcing a comma-heavy style, you can omit it.