The Toffees currently sit 13th in the table having largely recovered from their poor start to the campaign, with their disappointing 2-1 defeat at home to Arsenal on Sunday marking their first loss in eight league games.
Moyes, who took the helm at Goodison Park nine years ago, believes his squad is currently in its best shape it has been since his arrival on Merseyside.
And the manager wants the Toffees to demonstrate that talent by battling it out at the top of the table, insisting a place in Europe at the end of the season is his side's minimum target.
"As a manager - and I've been here for close on nine years - I find myself getting harder to please," said Moyes.
"The character we have within the dressing room is unquestionable, our football has improved, our style has been changing and we are moving forward year in, year out.
"Our targets have to be attainable and I strongly believe finishing in a European place is attainable for Everton. If I am harder to please then that's what we should be aiming for."
Everton found themselves briefly in the basement of the standings earlier in the campaign, but have turned their results around to suggest they can compete this season.
Moyes' confidence does not appear diminished by the Gunners defeat, but the manager admits he would like to see his side hitting the back of the net more frequently.
"We have lads who are determined to do well for the club and for a manager it's all you can ask, that your players are 100 per cent committed," he said.
"We need to start taking more of the chances we create, and all we can do is continue to make the right decisions, and be courageous enough to take the ball and pass it.
"We hope if we do that our conversion ratio will turn around and the opportunities we create will turn into goals.
"What we have to do is try to find solutions for us to score more goals and we will do this by looking within out own squad to see if there are any options we have missed."