Are the oceans rising or the continents going down? How can we know?
To the best of our knowledge, sea-level is rising because the volume of water is increasing.

There is substantial local variation in sea-level change; it's falling in some parts of Canada. But of the dozens of controls on local and global sea-level, the net effect is currently an average global rise of about 3&nbsp;mm/y. It's not really 'a riddle'. The phenomenon is well documented and there isn't too much mystery about the various causes. One could certainly debate their relative contributions.

Satellites already contribute much of the data that reveals the sea-level rise (and the relative stasis of the continents), and that substantiates some of the causes. For example, the TOPEX and Jason-2 satellites, constrained by gauges on earth, are important sources of data.

Two prominent causes of sea-level rise are thermal expansion and ice loss (for example in Greenland), but there are lots of others. Tectonic plates do definitely move up and down — it's an important phenomenon in geology and geomorphology, and it does affect sea-level in all sorts of ways (see that link).