What are you writing about?
In order to increase the traffic to you website, blog or article, you should have categorized your work and know who is looking for this information. Are you writing about recent events happening everyday or technology news? Are you writing very funny or very useful stuff that many people would benefit from? Or, are you writing evergreen content that will continue to be useful over the years and there will always be people looking for it?
SEO isn't that hard to learn, but for some niches and some keywords, the market is very saturated and competition is massive. In this case you have to either use social media or focus or other long tail less competitive keywords.
When to use social media?
Have you tried joining social media or social bookmarking sites? How are the results so far? Have you noticed some articles or posts getting more expousure in social media than others?
It is better to have abalanced approach towards using social media, you should try them but without wasting all your time and effort on them.
Social media loves WOW content!
Most of the stuff people share on social media are of the WOW type content! Ask yourself this
How to use social media
It is about socializing and connecting with people. Don't just go there and throw some links at the page then leave! If you want to spread your content, you should make friends, comment on people's posts and give them "likes" and they will do the same for you. Search for people who share similar interests with you. Engage and participate in the comunity, don't just use it to spam or advertise your content. Focus on a few websites only and build good networks in each one.
Other types of content
Some things just don't get much exposure in social media. For example, many people would share an article about "Top ten funniest baby videos" (wow content!) but only a few would share a page about "French grammar". The latter receives thousands of searches every month but how many would share it through social media?
Content, content, content!
Whatever method you use to promote your content, you must have quality content that is worth reading and sharing. It must be worthy sharing for social media sites to like it and the same applies for search engines to love it! Yes, search engines cannot read but they can analyse the bounce rate (do visitors keep browsng your site or just leave it within seconds), backlinks, the comments and interaction this page provokes as well as other factors. I'm not claiming to know what algorithsms search engines use, this is just me guessing. Some may argue, there is a lot of low quality content that make it to search engine results first page! I agree, but why would you want to risk being penalised or punished when search engines discover this content?