My boyfriend's mum has two house rabbits. She had a male first, but had a hard time finding another rabbit to accompany him (he didn't like any of them, and you can't have just one because they get lonely). They don't seem to do much other than lie around and chew things, but occasionally they'll run around and jump and stuff. Sometimes the male will thumb his hind legs on the ground if you piss him off (usually by walking near him if he's in a ****** mood).
They also poop everywhere. They have a special box they're trained to go in, but they go everywhere anyway. It's horrible.
I mean they all act differently obviously, but mine tend to keep to themselves for the most part sometimes they'll jump on the couch and hang out and they always let me pet them but other than that they just run around the house or flop over. Pic of my boy and girl they're both fixed.
I haven't owned a dog since I was a kid so I can't really compare, however my rabbits tend to be quite the handful when it comes to chewing on things wires especially, lost a few hundred dollars because they managed to chew my headphone wire. However I wasn't really that prepared to have rabbits it was kind of spur of the moment since they were going to put down this little guy and I couldn't say no to that face. Anyway they can be a handful you would need to bunny-proof your house if you plan on letting them run around a lot, they also chewed the molding on the bottom of my walls, I obviously adore these little guys but I wouldn't recommend getting them if you don't have a lot of time to spare also don't think they would make very good pets for kids.