There is no substitute for thorough testing. Do not cut back on testing, it has never been worth it. Plan to test, fix and re-test. Allocate someone to do mini tests during development in an effort to fix problems before they get too big. Being a tester requires skills. The tester needs to understand the role of the component being tested. The tester needs to be able to visualise everything that the component might be asked to do. Developers often do not make good testers, just as authors have proof readers, developers need a new set of eyes to spot the obvious.
Testing needs to be planned just like any other part of the project. Plan to test as much of the system as possible, preferably everything. If you are implementing a package, there may be some parts you can test less thoroughly, but that clearly has risks. Do not just test for one type of data, for example if you are implementing a real estate / property management system, test on industrial and retail and commercial and residential, test for single tenant buildings, test for multi tenant buildings, and so on. You may need to draw up a big grid of functions/features(x) and data types(y) so that you do not loose track of the testing status. Its a good idea to keep in close contact with trainers and developers during testing, sometimes perceived errors are in fact being triggered by a wrong use of the system. These problems still need to be fixed, but their impact may change from critical to simply irritating. It is also possible for serious errors to get hidden by presumed user errors.
Some users will struggle with testing, because they will abandon hope after the first issue. From one viewpoint you could exclude these users from doing testing, as they offer little value to the testing process. But it is precisely these users who are most likely to struggle later. This type of user tends to learn more slowly, so it is probably better to involve them, despite the hand holding required, as they need the experience and they will need to use the system later. These users need to get past their difficulties with the system as soon as possible.