The West Indies enjoyed back-to-back wins by five wickets and 98 runs against the tourists at the Antigua Recreation Ground on Saturday and Sunday but Lara, in his third spell as skipper, wants to see bigger contributions from his batting line-up.
"We lost too many wickets on Saturday chasing 152, and on Sunday, we were almost bowled out," said Lara.
"We have to start thinking about 300 runs-plus for the remainder of the series and then I'll definitely be happy about it.
"I think we have been a little slow off the mark because it's our first couple of games back after the trip to New Zealand and you can tell by the way that some of the guys have been playing.
"They are still feeling their way, but I am sure that by the time the series resumes next weekend in Guyana, we are going to have all the guys pumped up and really going."
Lara's decision to drop down the order to six has been the topic of some debate, but he indicated it was part of his side's tactics to expose some of the players in preparation for the upcoming series against India next month.
"The top four or five batsmen should dominate the batting in these 50-over games," said Lara, who celebrates his 37th birthday on Tuesday.
"If this takes place, the lower order batsmen will not get a knock, so I am trying to get some of the guys lower down the order to get a knock."
Zimbabwe captain Terrence Duffin acknowledged his side were on a learning curve and expects to see them improve by the time the series is over.
"After getting a few early wickets, I think we did well to keep them down to 242," he said.
"We lost too many wickets too early.
"We've got a few days before the next two matches, and it gives us some time to go and polish up."
The series continues on May 6 and 7 at Georgetown's Bourda Oval when the two teams contest back-to-back matches.
They will also play the first Limited-overs International under lights in the Caribbean at Gros Islet's Beausejour Stadium on May 10 before the series ends on May 13 and 14 at Port of Spain's Queen's Park Oval with a double-header.