India Test batsman Pujara made 112 on his Nottinghamshire home debut, sharing a 185-run stand with Lumb (112) for the fourth wicket at Trent Bridge.
Lumb fell six overs before the close following a 204-ball stay at the crease and after the hosts had been put in.
Craig Miles (3-88) was the only bowler to prosper in the Nottingham sunshine.
Nottinghamshire are currently unbeaten this season and lead the Division Two table.
Centuries from Michael Lumb and Cheteshwar Pujara put Nottinghamshire in the commanding position of 357-5 on the first day against Gloucestershire.