Your second example is unusual - we don't usually write out the full form 'the 24th of January'. We say it, but don't tend to write it often. Basically, there are four ways of writing a date, with the number before or after the month, with or without the 'th'
Whether or not it's written, we always use the ordinal number (first, second and so on, or in this case, 'twenty-fourth')when we say the date.
Speakers of American English favour the 'month first' format, and would typically say 'January twenty-fourth'.
British English users tend to add the 'the', and say 'January the 24th' or - if the date is written first - 'the 24th of January.'
You can choose which ever you like.