Can someone, without any prior knowledge of philosophy, pick up Hume's A Treatise of Human Nature and read it without a problem?
Yes, of course one can start with Hume's Treatise. I recommend the Introduction, where Hume states the basis of his philosophical method:
And as science of man is the only solid foundation for the other sciences, so the only solid foundation we can give to this science itself must be laid on experience and observation.
I consider this view a big step ahead taking leave from Christian philosophy. On the other hand, I would criticize this view because of the importance of sciences like physics or cosmology for today's world view.
Sections 1, 4, 6 from the Treatise contains Hume's revolutionary view on the human person as a bundle of perceptions:
But setting aside some metaphysicians of this kind, I may venture to affirm of the rest of mankind, that they are nothing but a bundle or collection of different perceptions, which succeed each other with an inconceivable rapidity, and are in a perpetual flux and movement.
One can ask how this view relates to current views from neuroscience. In any case, Hume's view is again a farewell to the traditional concept of a human soul.
The Treatise is a ponderous tome. Hence you will probably like to read only selected passages. A secondary source like the book recommended by Mauro is helpful.
As a companion to the Treatise I would like to recommmend Hume's An Enquiry Concerning Human Understanding. The latter contains in chapter VIII Hume's famous discussion of the relation between actions of will and causality:
[...] men still entertain a strong propensity to believe that they penetrate farther into the powers of nature, and perceive something like a necessary connexion between the cause and the effect. When again they turn their reflections towards the operations of their own minds, and feel no such connexion of the motive and the action; they are thence apt to suppose, that there is a difference between the effects which result from material force, and those which arise from thought and intelligence. But being once convinced that we know nothing farther of causation of any kind than merely the constant conjunction of objects, and the consequent inference of the mind from one to another, and finding that these two circumstances are universally allowed to have place in voluntary actions; we may be more easily led to own the same necessity common to all causes.
Yes, but you'd likely get more out of it if you're familiar with some of the philosophical positions he's challenging. This however, goes for most philosophy.
Also, if you're concerned about the readability of the volume, you can find an abridged version written in more modern language here.
As an aside, the site above has many other classic philosophers (including Kant!) rendered in more readable, less weighty prose.