Hi, and welcome to the forums.
I don't want to sound derisive, but the Badger 250 you have chosen will only be marginally better than using a spray can. Its advantage is that it gives you access to a whole range of colours which are not available in spray can form. however, if that's what you have, we'll just have to make the best of it. ![Smile [:)]](/emoticons/icon_smile.gif)
For wide-area/single colour applications, it will do just fine. For multi-colour work, you will have to learn the art of masking.
You can use pretty much any sort of paint as long as it's suitably reduced (thinned) to spray smoothly. I'm not sure if an adaptor is available to mount alternative paint jars on a 250, but I doubt it. You will need to mix your paint/thinner and pour it into the supplied jar and then screw the jar onto the "brush". There are many ways in which you can extract paint from a jar without making a mess, including squeeze pipettes, eyedroppers, small syringes (without the needle of course).
I'd suggest using the recommended thinner for your acrylics - though most are "water based" they don't necessarily thin very well with it. Some, like Tamiya, you can use alcohol, but the same alcohol won't work with others (eg Vallejo - which can be thinned with distilled water)
IF you are using enamels, you can thin your paint and clean your brush with mineral spirits, available from your hardware store at a much lower cost than the "hobby brand" thinners. When using acrylics, you can clean your brush by flushing with water and then using alcohol or Windex (blue window cleaner containing ammonia) If using Windex, be sure to rinse well afterwards.
Whilst there are some paints which are said to be "airbrush ready", though tey may be convenient, they are lower value for money as the equivalent "regular" paints (even from the same manufacturer) require thinning for airbrush usage, effectively giving you a higher volume of paint (after thinning) for the same price.