What's the best compromise of higher visibility of the beam and burning capabilities? Well, the greater the power, the better the both.
But if you mean you want the lowest output laser that you can both visibly see the beam and burn stuff, I'd go with a 50mW green focusable laser, or a 100mW ordinary green laser. For other colors, you'll need slightly more power to see the beam more clearly.
well, you dont need to compromise between visibility and burning ability, since they're proportional to each other (via power). unless you mean about how blu-ray tends to burn some stuff better. green is best visibility anyway. i'd recommend o-like or rayfoss for a starting laser. anything from around 100mW will burn with green. they tend to be focused at infinity so if you want a laser just for burning then a focusable red/violet would be best, unless you wanna spend a bit more for a 445nm build.
Remember too that before power even increases to levels capable of burning, you need glasses to safely use the thing. Safety glasses in turn reduce the visibility. Dreadful catch 22.
Some of it depends on where you live too. Here in Louisiana with 1 billion % humidity even a 5mw green can be seen outside at night.