Don't move too quickly OR too slowly
This may sound like a big "duh!" but I'll say it anyway: When it comes to selecting and implementing a new AMS, don't move too quickly or too slowly.
If you move too quickly, you are very likely to overlook things. During the selection process, this means overlooking functionality that you will need but hadn't thought to ask about. I've worked with clients who have come to me after selecting a system that they found out could not function the way they needed it to.
Moving too quickly during implementation manifests itself in taking things "live" before they've been thoroughly tested. And as I always tell my clients, either the staff tests before go-live or the customers will test after go-live.
Moving too slowly also has drawbacks. Moving too slowly during the selection process, especially when comparing multiple systems, can make it very difficult to remember which system does what, what systems look like, and so on.
During implementation, moving too slowly can result in a lack of focus by staff, dramatically increased budgets, and frustration from all parties.
What you're looking for is the "Goldilocks" speed. Fast enough that things don't bog down, but slow enough that avoidable mistakes aren't made. As a general rule of thumb, the selection process should be completed in three to four months, and the implementation process between six months and one year. Of course, this may vary dramatically for certain situations, but these are good guidelines to start with.
Remember: Not too quickly, but not too slowly!