daefothepast has good tips! (And as an example, here's what my update tweets look like!)
I also recommend following other comic artists who are doing the sort of thing you do! Occaaasionally people will follow back, but the real reason this is helpful is that you can see how other folks promote their own work, and learn what works and what doesn't by watching them! You can also see the sorts of things other creators are talking about and have topics to bounce off of in your own tweets.
Similar to that, there are still a few hashtag events that happen -- for example, webcomicchat still runs a discussion on Saturdays and Sundays, where for an hour each day they tweet out questions on a particular topic, and creators chime in to share their experiences, opinions, and advice using that hashtag, and it's shared with everyone who's watching the hashtag! In the past it's been a really good way to get involved in the comics community, find creators with insight that you're interested in, and sometimes get noticed by others!
Tumblr can be a bit more difficult; it can feel like shouting into the void for a while until you find people, and I don't think tumblr works well if you're Intentionally Promoting Your Product unless you already have a bit of a following. Think of it this way -- if you see someone's blog, and they don't post about anything except an indie comic that you don't read, then why would you follow them unless you're already reading the comic? So it's hard to attract new people that way! To me, tumblr works best if you do things outside of your comic ALSO, whether it's a community you're involved in, or you have a ton of OCs you like to draw and talk about, or a fandom you're really into and like to draw fanart for -- these things mean there's a reason for people to follow you even if they don't know your work already. I usually describe the difference as, Twitter is a business convention, Tumblr is a party -- if you go to a party to hand out business cards, people won't be interested. But if you go to a party and start chatting with folks about their favourite cartoons, you have a chance to make some connections.
In general, though, social media is something that's constantly changing. What works right now might not work in a year. What worked for me 4 years ago might not work now. So I think the best way to approach any social media is to find and follow the people who are using it well, and watch what they're doing! Heck, I'm still learning new things all the time by watching what my peers are doing!