We've gone through a very similar process over the last few years and I can share some things that have worked for us.
What to Test
There are a lot of facets to web apps and there's a lot of benefit in testing behind-the-scenes code, but the best place to start is usually the web interface (so browser automation tests). This will help you identify where problems exist, but may not offer much insight in why those problems are occurring.
Important Types of Tests
The positive test (it does what it says it should) is the obvious one. Negative tests (bad inputs and invalid selections, wrong logins, attempting to access resources the user doesn't have access to) are the next set of tests you'll want to add into your test plan for testing the interface. Your QA person and developers should definitely collaborate on brainstorming these. The QA engineer will think of things that the developer is too close to the project to come up with, but the developer will have insight into how the code works to know where snags might occur.
The natural inclination is to test functions, which can be good, but it's also important to test full processes. A number of features that work fine by themselves but don't integrate to each other properly can be disasterous.
Test Automation and Regression
The more sections you add to your application, the more time and effort it will take to test the full site. It sounds like you're just testing what you think changed, which is pretty common in those scenarios. The nice thing about automated tests is that you can simply add your new tests for a new section to the existing suite of automated tests. That way the role of the QA Engineer is to build those tests and the automation can go back through the entire suite of tests to ensure that other functions were not broken by your new changes.
For automation tools, the two I'm partial to are Ranorex and Selenium. I think Ranorex helps you through building the test more, but is still incredibly flexible because it's all backed with C# and you can directly edit the code. The downside is licensing. Selenium is free and very poserful, but requires a little more effort to implement. Also, Selenium has a few gotchas, like if an element is hidden, Selenium can still click it, even though a real user couldn't.
I know Ranorex does multiple browsers and I think Selenium does too.
I hope this is helpful to get you started.