このようなものはどうですか:
^([a-zA-Z][a-zA-Z0-9_]{3,})$
パターン全体を説明するには:
^ = Makes sure that the first pattern in brackets is at the beginning
() = puts the entire pattern in a group in case you need to pull it out and not just validate
a-zA-Z0-9_ = matches your character allowances
$ = Makes sure that this must be the entire line
{3,} = Makes sure there are a minimum of 3 characters.
You can add a number after the comma for a character limit max
You could also use a +, which would merely enforce at least one character match the second pattern. A * would not enforce any lengths