This looks like an x-files post but it isn't. While training today I kept telling the trainee that he didn't know his employees yet he was going to be relying on them to get things for him and do things for him and that his name would ultimately be on lots of things that he hadn't actually done himself. He'd be standing on their shoulders. He was a supervisor now, and as a member of management he had to work hard on making sure his employees gave him reliable information, reliable results, trustworthy results.
The only way he could do this was to check what they do. He had to give assignments, ask questions, and follow up on both. If an assignment was important enough to assign it was important enough to follow up on. If you're not going to follow up on an assignment you shouldn't give it. Your employees' time is valuable to them. Don't waste it on things you're not going to follow up on.
So. When an assignment is given, a project assigned, follow up on it. Call the next day. Check on the thing next time you go through and comment on how nicely it was done or what could be done differently etc. Let the employee know that their work is/was important to you and that you appreciate it. The constant checking up on it and them. The follow up on assignments and positive reinforcement when they do it right will create a situation where if you're out of town or away and can't get there you know they're doing it to your expectations.
Until they get to that point though follow up... check up... and make sure that your expectations are explained, made clear, and followed up on. Don't just give the assignment and walk away without checking up or following up. Don't just trust it was done. Not because you don't trust them... but because you want to make sure they can learn to trust you. Trust you as the new supervisor/manager to only give assignments that are valuable. Trust you to not waste their time with busy work. Trust you to appreciate the work they do.