This is a really tough question to answer because the art world is very fickle; plus the type of artwork is so varied. I am definitely not an expert in this area, so I am only offering my opinion.
I have met tattoo artists that sell their art on t-shirts. I know of artists that sell thousands of dollars worth of original art on eBay. Also, there's that special relationship between an artist and their gallery that shouldn't be undermined.
I would recommend taking a look at the different types of products produced by your art -- whether it's originals, giclees, prints, t-shirts, or digital forms -- and ask yourself these questions:
- What is my target audience for my type of art?
- Where will my target audience typically shop for my type of art?
- What are other, similar artists doing to sell their art?
Then, I would recommend testing your ideas, always looking for feedback.
If you are working with a gallery, I would recommend offering them a product line (e.g. a series of paintings) that can only be purchased exclusively through them. Then if you choose to sell other art online, you aren't undermining their efforts to sell your artwork.
Selling art in general is a difficult task. It depends on so many factors starting with the actual art -- whether it's likeable, etc. -- and ending with the price point. People could love your art, but not be able to afford it; which brings us back to the importance of feedback.