This is an idiomatic usage which might be more commonly expressed as to go a long way toward sth. The toward sth prepositional phrase is omitted here, but toward some kind of success is implied. The American Heritage Dictionary of Idioms definition states
Have considerable effect or influence on. For example, This argument goes a long way toward proving the scientists are wrong.
ODO offers its definition under go:
1.8 Used to indicate how many people a supply of food, money, or another resource is sufficient for or how much can be achieved using it:
the sale will go a long way toward easing the huge debt burden
a little luck can go a long way
A popular formulation is to say a little [of something] goes a long way, meaning a small amount of something has a greater effect than one might expect. A little kindness goes a long way means that showing a small amount of kindness repays itself manifold in the kindness received. Or, a cookbook might caution that a little ghost pepper goes a long way, meaning that a small amount of ghost pepper contributes greatly to the spiciness— in other words, don't use too much!
To be more explicit, the sentence could be written as
Caution and common sense contribute greatly towards success when working with computers; problems are often much easier to fix than they appear.
Caution and common sense will be very useful when working with computers; problems are often much easier to fix than they appear.
It can also be expressed using other words for distance, as in go far, to go miles towards sth:
Hoffmans' deep and lasting sympathy for his characters goes far toward explaining his particular genius as an actor. [American Film Magazine]
Something as simple as [good packaging] goes miles towards making it more enjoyable to bring a lunch. [Lifehacker Australa]
For an older related thread at EL&U, see “Go a long way to” + gerund vs infinitive.