Everton travel to Anfield on Saturday for their first derby of the season.
Moyes began by admitting he's been pleased with how the team has gelled with the new arrivals signed over the summer.
"Well, that was the plan," he began, "and we always hoped to get off to a good start.
"We aimed to be near the top of the league at this stage and it's now given us something to hold onto and fight for.
"Look, we didn't get all the players we wanted in the window, but most managers would say that. But I must say we picked up a fair amount and we did a lot of good work over the summer, so hopefully we're moving in the right direction."
He continued: "I've told the players that we want to go away from the position we have been in for the last few years, it's been difficult for the managers, it's been difficult for the club and the players especially over the last few years.
"We want to enjoy it and we want to try and play better and win more often and we are trying desperately to challenge the big teams and we will try and do that throughout the season."
Liverpool always difficult over the years
On facing Liverpool, Moyes continued: "Well, any derby in a big city is always important to the fans.
"I must say my time here facing Liverpool has always been difficult. We've faced some great Liverpool teams with some great players over the years.
"But we've always been competitive and that won't change. We will be trying to find a victory."
Moyes also said: "I think the gap between the two clubs was as close as it has been in my first stint.
"But then we had to look after ourselves as they began winning trophies. Though now, I must say we're heading to a much better place.
"Obviously, they've bought some really top players under Arne Slot. But we've improved too. Undoubtedly, they have some very good players - including (Alexander) Isak."
Moyes was also asked about whether his new players understand the importance of the game.
He said, "The job for the players is they know it's very important to win a derby. It's come quite early this one, so it will be a good chance to see where we are."
On Liverpool and their stuttering form, Moyes also stated, "Well, that's a good question.
"Liverpool have shown at times they've been very good, but at other times have shown they can be vulnerable and concede goals. And that's something we'll be looking at."
Garner's form
Moyes highlighted the form and contribution of James Garner so far this season.
"Yes, Leighton Baines has had an influence with Jimmy playing at left-back at times. But we all have an input.
"And I say I think Jimmy is playing as good as any English player at this moment. Certainly in terms of midfielders, he's been as good as any in the league."