Put into bat on a damp, cloudy morning, Glamorgan reached 256 for five before bad light brought an early end to the day with 5.2 overs remaining.
North scored 68 and Goodwin 57 to turn things round for Glamorgan after they slipped to 98 for three. Nathan Buck and Ollie Freckingham both took two wickets for Leicestershire.
With Ramnaresh Sarwan joining up with the West Indies squad for the Champions Trophy, Australian Joe Burns made his Leicestershire debut as his replacement, while Josh Cobb took over as captain.
After bad light delayed the start by 30 minutes, Cobb won the toss and decided to field. He was rewarded with two early wickets both snapped up by Freckingham. He had William Bragg lbw for five in his third over of the morning and then, having seen Stewart Walters twice dropped in the slips, made it third time lucky with Michael Thornely clinging onto the catch.
It was an impressive spell by Freckingham, but Ben Wright and North weathered the early difficulties to share a third wicket stand of 59. Both batsmen were quick to seize on anything loose and the boundaries began to flow.
Wright hit seven fours in his 49, but then edged a catch behind off Buck, making his first appearance of the season after recovering from an ankle injury.
But a partnership of 86 between North and Goodwin regained the initiative for Glamorgan. Goodwin was the first to his 50 which came off 56 balls and contained 10 boundaries and North soon followed, reaching his half century off 133 balls with nine fours.
Buck finally made the breakthrough by trapping Goodwin lbw with the last ball before tea, and then Anthony Ireland, making his Leicestershire debut, had North caught at short extra cover off a loose drive for 68.
Bad light sent the players off the field again and at 6pm, although they returned 25 minutes later, only four more balls were bowled before play was called off for the day, with Jim Allenby unbeaten on 39 and Mark Wallace 12 not out.