Where are all these imaginary people who have ample time and inclination to discover you? I can’t imagine a single gallerist who, after curating, framing, installing, promoting, and selling a show — in addition to managing a staff, courting new collectors and arts patrons, and maybe even having a home life in the odd moment outside of work — who is going to be able to wade through the deep seas of information put out there by artists who are promoting their work, in order to find you. I hope I am making this point clear for artists who sit in their studios hoping to be “discovered”: when you hear the phone not ringing, it will be the galleries not calling.
But you don’t have to take my word for it. I wrote to several arts professionals about being found and they all said the same thing: it doesn’t work like that. One gallerist from Boston said, “I would agree with you on the ‘discovery’ idea. With all the ‘noise’ in our contemporary culture, subtlety when it comes to self-promotion is a losing idea. Years ago, before social media, [another] gallerist told me he had a rule of three: if he heard the same name from three different people he would check out the work. The more ‘buzz’ around the work the more attention it is going to get.” A gallerist in San Francisco told me, “I couldn’t agree more. An emerging artist should be doing as much as he/she can to connect with others and find exhibition opportunities. So much can be learned from showing and working with others. This is how an artist builds his/her community within the art world and an important audience for their work. From a marketing or sales perspective, it’s of course far easier for a gallery to work with an artist who has already established a positive reputation.”
As for “playing it subtly” and trying to meet the “right people,” those are strategies likely to disappoint unless you already know influential arts professionals who have previously indicated that they are happy to promote your work. Here’s where you need to be honest about your ability to get within shouting distance of whomever these “right people” might be — I’m guessing that you’re not currently rubbing elbows with a bunch of gallerists and collectors who summer in the Hamptons. In the long run it’s better to rely on the quality of your work and your marketing than to try to strong-arm friendships with the mighty. I’d rather see you take a more sensible route, one that will keep you from dying with a note to David Zwirner clutched in your paint-stained hand.

Philippe Parreno, From November 5th Until They Fell Down, 2013.
Since your ultimate goal is to get some attention from galleries, you’re going to have to start elsewhere and build up that “buzz” that our friend in Boston is talking about. Have your work photographed by a professional. Make it easy for people to find you by having a website with a good layout and visible contact info. Email a few images and a short press release to online and print publications likely to be interested in your work. Make a Facebook page, a tumblr, or an instagram account. Contact the local papers. Sometimes you can get other groups interested—for example, if you paint trains you could send an email to the National Model Railroad Society. Don’t laugh! It’s a somewhat-facetious example but I do know artists who have made sales and built a buzz on the basis of their work’s subject matter, so brainstorm people and organizations who might connect to what you’re doing. Don’t discount any avenue you have for creating a dialogue around the work.
While you’re doing all that, line up some studio visits with arts professionals — gallerists, independent curators, even arts writers. Do your homework and figure out who might be interested in your paintings. If you’re not used to doing studio visits, practice by swapping visits with art friends and/or ask a few former professors to come for a short visit so you can “warm up” and be ready for when a gallerist arrives.