If you are doing it to complete the whole stage then park up anywhere around the city centre and off you go, with the option of getting the train back from Harrogate to Leeds (not Headingley).
Personally I wouldn’t bother with the train and would add the Harrogate to Leeds ride to the end of the stage. It’s easy riding, and has that ‘downhill’ feel to it as you ride back to Leeds.
If you only want to do the ‘scenic’ parts of the rides then miss out the Leeds exit and start at Harewood as suggested above, and complete the loop with a Harrogate to Harewood ride. You would be missing out on perhaps 7-8 miles of getting out of Leeds on a section that doesn’t have many redeeming features, and make the miles up with the Harrogate to Harewood ride.
As for the ride out of Leeds up the A61. It’s fine. Urban dual carriageway with a few roundabouts. Just the usual traffic to be aware of – no really high speed motoring or HGVs of note to worry about.