Do good tires lessen the chances of your innertube bursting?
Its extremely unlikely for an inner tube to just burst.

It sounds like you are suffering from punctures.
There are two types of puncture; the first is an object penetrating the tire and inner tube, and the second is a 'pinch' puncture where an impact causes the inner tube to be pinched between the tire and rim.

Many bikes (even expensive ones) come with poor quality tires as standard, and yes, changing them can make a big difference to penetration punctures.
No tire is 100% puncture proof, but something like a Schwalbe Marathon Plus is very close, but is heavy and can give a harsh ride.
Continental Gatorskins are favored by many as a good compromise between puncture resistance, rolling speed and comfort.

Pinch punctures are usually caused by hitting an object (such as a pothole) with too little pressure in the tire.  Adding more air to the tire (don't go over the maximum limit) usually prevents these.