I'm a very regular Tumblr user and I always check if artists here have a Tumblr where they share their work.
So here's a list of advices that you might want to follow when you use Tumblr to share your art:
1. Make a blog exclusively for your work.
I don't care what you think: if you join tumblr you are going to find amazing things that you will want to reblog. So make a blog for your work and another one for your fandoms or inspirations (you can have all the blogs you want, maybe create the account with a blog for inspiration where you can follow others and then create the blog just for your art). It's a bit disappointing to follow an artist on tumblr and see everything except their work or having to go through a million posts before you find the exact thing you wanted to reblog.
2. Use the tags!
That's how Tumblr works. Make your art known, use the tags. But don't go crazy on them. The tags that are more important are the first 5 so use main descriptive tags like Portrait, Illustration, Black and White, Landscape, Art, Design, Crafts, Animals, etc. . And, the most important, the one that you should use always: Artists on Tumblr</b>. After that, you can use as many other tags as you want.
If you use tags correctly, you get the chance to be feaured and that will give your are great exposure.
(Another reason to make a blog only for your works).
3. Update your blog.
I know it might mean a lot of work, but it's a bit sad to see some tumblrs left to oblivion and see that the artist is still producing loads of new stuff.
4. Let deviantart browsers know you have a blog in Tumblr.
I've seen some of you posting that Tumblr link under each deviation, that's great. Most of the time I have to go to each profile and look. Sometimes I find the link in some old journal entry.
The best I've seen: when they post the link to Tumblr on each of the deviations. Yes, the exact link. So if someone wants to share it, they just have to click and reblog it from you.
I don't know why, a lot of people on Tumblr like to delete the credits from the posts they reblog. It's rude and annoying, I know, but it will happen, specially if you make a long list instead of one line. What you can do is set the click through link to your website or your deviantart page because no one will be able to change it. If you just use the content source link, some nasty rebloggers might change it.
I think this is all for now.
Hope it helps!