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Great question. I can't comment on QA training programs, but I can tell you that any experience with QA / testing is going to put you in better position than having no experience.
One way to get experience is to seek out part-time contract work on sites like https://www.upwork.com
You could also try https://www.utest.com
Without experience, your best bet is to look for simpler projects first that just need someone to run "manual testing."
The two main steps in manual testing are:
1) Write a set of test cases. These are very detailed specs that describe:
a) What actions are you taking in the application? E.g. "Click the sign up button"
b) What is the expected state of the application afterwards? E.g. "The registration form is displayed"
The cases are normally built based on a combination of product specifications and experience with using the app. Sometimes these cases just live in a spreadsheet; other times they live in a tool for QA.
2) Run through each manual test on a "release candidate" and record you results. A release candidate is simply a version of the app that *might* be ready for to go live but hasn't been thoroughly tested yet. This step gets repeated until a viable release candidate has been verified.
Now, having said that, let me also tell you that the best QA people are the ones who can also write some code. Being able to code a bit not only enables you to automate more of your work, but it also gives you a better insight into what you're really testing for.
Tags: The recurse center, aka hacker school
Tags: Quinnipiac university