The journey from Leeds to Grimsby, by rail, usually involves taking a train bound for London Kings Cross and then changing trains at Doncaster. (It takes about 30 minutes to Doncaster and then about an hour from Doncaster to Grimsby). Full timetable details are here: http://www.nationalrail.co.uk/
you've not said when you're travelling - if imminently, be aware that due to recent flooding the line between Doncaster and Scunthorpe/Grimsby is currently closed. (Buses are running instead).