The phrase "Just bought an airbrush" is the key. Nobody I know of ever bought an airbrush for the first time, took it out of the box, and started spraying fantastic paint work. They take some time and practice to become proficient with. An airbrush is a tool, nothing more and nothing less. The knowledge and ability to use one properly comes from the hand holding it, not the tool itself. After all, you didn't purchase your first paintbrush and immediately start painting as well as you do today, did you? If you want to use an airbrush you'll have to take the time and invest the practice to learn to use it properly. There are no shortcuts.
Acrylics typically bond very well to Tamiya spray laquer. I use that spray frequently as a primer and once it has cured have never had problems with acrylics (or enamels or laquers) not bonding to it properly. If you are handbrushing modeling acrylics (especially Tamiya acrylics) they are very difficult to hand brush because they dry so quickly. In most cases the first pass is halfway dry before you finish making it, and subsequent passes just cause it to roll up and look rough. Try using some acrylic retarder (available at any art supply store) to slow the drying time down. You can also use Tamiya thinner which already has retarder added.