You're heading in the right direction!
There is a whole website/blog that talks about using small portable lighting gear for everyday use and getting the most out of simple elements. The "Strobist" site has many resources for what kind of kit to get and how to use it.
Here's the deep link for the basic kit that they say you'd do well to get for the starting shots: Strobist: Lighting 101: An Ideal Beginner's Lighting Kit.
For my purposes, I started with some off camera triggers (optical triggers are fine, too), two flashes, two stands, one umbrella, and some gels. The rest of the diffusers, reflectors, and snoots/grids came from cobbled together bits from around the house. I use foam craft paper to act as larger bounce cards and flags.
For the rest of your questions about mixing continuous lighting with strobes, you just need to understand how light works with blending the short impulse with ambient light. This balance can be learned by trial and error as well as through some exercises. In the end, it's all about ISO, F stop, shutter speed. Each one controls the way that light comes into the camera and how the camera responds to it. In short, it does make sense to mix, so long as you know the color balance of your lighting and can control it. To that point, be prepared to gel your flash to match the color of your ambient light.
What I'd also suggest is getting together with other local photogs to practice settings and borrow gear. That will teach a lot about what effects modifiers may have and how to best use them. I've started going to a local meetup group and have learned tons in just a short while.