There are lots of different types of sea sponges, some look like the sponges you may use in the bath, others like like blobs of muck, some of the sponges that are washed on the shore are the exoskeletons of the sponge after the sponge itself has died.
The Queensland Museum has some excellent information about sponges. You can find it here.
I've pinched snippets of information for you -
"Many species of sponges are actually capable of some locomotion, just very slowly. ...sponge larvae are highly mobile, capable of crawling across the sea bed, but generally adult sponges are firmly attached to the seabed."
"Many or most sponges reproduce sexually at some stage of their life cycle, producing larvae that can disperse and recolonise new habitats. These larvae are either incubated within the parent and born live (vivipary), or the eggs and sperm (gametes) are broadcast directly into the seawater where fertilisation takes place (ovipary). Oviparous sponges may have separate sexes (gonochoric), or may be capable of changing sex, from male to female and back again, in sequence (sequencially hermaphroditic)."
Asexual reproduction also occurs in sponges with some budding and others fragmenting to produces new sponges (little clones).
Next time you're having a bath with a sea sponge...you can wonder about its previous sex life :)