I think we should reach for mammalian and human-like AI first. Build better hardware which can run nn networks the size of our neocortex+thalamus, some optimizations/hacks are allowed, but it should still embody the same rough functionality. There's actually some interesting work being done in this direction, but it's a bit disappointing that there are only a few serious contenders to this race.
As for Strong AI, it could be anything, but it should be able to do everything human-like AGI can do and more (this doesn't mean just increased speed, that's included in human-like AI, but increased capacity/depth could be acceptable). Alternative designs (non-biological neural network-like) would also be interesting, but I don't think we'll be getting some usable ones too soon, and there's that whole FAI problem, which I'm ignoring as far as human-like AI ones is concerned (I could give more detailed reasoning here, but I'm too tired to it all, so I'll leave that for another time).