Both Keaton Jennings and Scott Borthwick were dismissed for ducks by Ryan Sidebottom before Will Smith was caught at third slip by Kane Williamson off Jack Brooks on 20.
Stoneman was joined by Ben Stokes (18 not out) in the middle as he reached his 50 off 70 balls - including nine boundaries - before adding 23 more runs to guide Durham on to 117 for three by lunch.
Chris Nash also passed the 50 mark as Sussex made it to the break without loss at Warwickshire.
Luke Wells scored 44no while Nash's unbeaten 54 that included eight fours saw the visitors go into lunch on 107 for nought after 30 overs at Edgbaston.
Nick Compton was the early casualty at Lord's as Somerset put on 101 for one against Middlesex.
There was an unbeaten 55 for Marcus Trescothick with Chris Jones adding 15 runs to the board but Compton was caught by Sam Robson off Neil Dexter having scored 31 off 71 balls.
In Division Two, there were two half-centurions at Lancashire as Ashwell Prince and Luis Reece guided the Red Rose to 133 for three against Hampshire.
Prince had put on 63 before Sean Ervine saw him caught by Adam Wheater but Reece remained in action having added 50 runs from 91 balls - including 10 fours.
Paul Horton had put on four runs before James Tomlinson got him out and Andrea Agathagelou had four unbeaten runs by the break.
Leicestershire lost three wickets to Worcesteshire for only 69 runs.
Grace Road saw Greg Smith (one), Niall O'Brien (31) and Shiv Thakor (nine) dismissed by Alan Richardson, Joe Leach and Shaaiq Choudry respectively while Ned Eckersley (21no) and Tom Wells (5no) contributed 26 runs between them.
Essex were struggling at the County Ground as Jaik Mickleburgh (26) and Gautam Gambhir (nought) fell to Graeme McCarter and Will Gidman, but Greg Smith (48no) was just shy of reaching his half-century alongside Owais Shah (11no).
Elsewhere, Andrew Hall and Steven Crook took the respective wickets of Glamorgan batsmen Gareth Rees (39) and Will Bragg (32) with Murray Goodwin (10no) and Chris Cooke (4no) adding 14 runs to give the visitors a 96-run lead at Northamptonshire.