It's far more productive to head to a well-connected position on the map - a tube station or something - so you've a better chance of reaching his position when he first rears his head than to try tracking down Mister X during your first few turns.
If Mister X plays a double move or obscures his route with a black ticket, it's safer to return to well-connected map locations rather than attempt a capture. That way, you're making it much harder for him to escape.
As Mister X, be sure to aim for well-connected areas when your position is about to be revealed. This makes it much harder for detectives to accurately predict your next move.