Hire people that want to a career in software quality assurance rather than those who see testing as a stepping stone to development. I do a lot of both, and good testers are often poor developers and vice versa, as the disciplines involves quite a different mind set. Check the links on this post over on SQAforums for some great resources on this topic.
Edit: As requested, some of my experiences in terms of getting the most out your existing resources (time / people / budget) to produce a quality software product.
Rather than trying to add various ideas together to improve your testing, develop a detailed testing strategy in conjunction with all of the stake holders. (developers, managers, testers, and users having to deal with bugs). Start by establishing your requirements, i.e. what do you hope to achieve by testing. Is it primarily finding new bugs, improving usability, is it managing risk to the extent that the software complies with the supporting documentation, help files, etc... is it having the confidence that making small changes to you app will not cause the whole thing to come falling down around your ears?
Having figured out what you want to achieve through testing, you then need to come up with a plan of how to do this. Have you looked at automated testing for regression, exploratory testing for new and undocumented features, etc... FWIW, I've outlined the goals of my testing strategy here.
You may also want to consider what QA procedures you have in place such that there are fewer bugs to start with, e.g. Unit testing, TDD, Design by Contract, static code analysis, etc... Too many here can be as bad as too few where resources are limited as by repeating unnecessary testing you are drawing resources from other necessary testing.
Here's the most painful one. Are you properly resourcing you testing effort? What is the ratio of testers to developers? Do you need to reduce the amount of money spent on development in order to increase the amount of money spent on testing? Once you have developed your testing strategy, implementing it will require an appropriate budget. Unfortunately, good QA does not come for free, although carefully planned it can provide huge return on investment. Don't forget any budgetary constraints when developing your testing strategy.
As you may have gathered, proper testing and SQA is a complex technical discipline, and should not be thought of as an adjunct to the end of software development.
One more Edit: I found it very beneficial to immerse myself for a while in SQAforums as you can ask questions of very helpful experts in the field. (FWIW, I'm just a dev who also faced a major QA crisis a couple of years back, I'm not a QA).
And one last edit:
What kinds of things do you find
yourself "teaching" testers?
I don't believe that developers should be teaching testers how to test, any more than testers should teach developers how to code. If anything, testers should be co-operating with developers to test efficiently and effectively. The 'education' goes both ways here.