If you really want precision. make a template of the top (and any other parts you want) out of 1/2" MDF. It shapes and sands wayyyyy easier than plywood. Spend a good amount of time getting your templates just right and you'll save that much time shaping your work pieces. In fact, even the 3/8" plywood may not be all that bad to shape. MDF is definitely the easiest though. Once your template is just right, do the 3/4" piece of plywood first.. Lay your template on the workpiece and trace it onto the plywood. Remove the template and use a jigsaw to cut away most of the waste, cutting about 1/8" away from the line. Lay the template back down and clamp it down this time (or use double sided tape; if you choose this method be careful; it's very easy to use too much tape and then you'll have a fun time getting the workpiece and template apart). Now use your router and a pattern bit (bearing on top) if your template is on top of the plywood or a flush trim bit (bearing on the bottom) if your template is on the bottom to trim away the waste. I prefer pattern bits. Since the bearing is on top there's less of a chance the bit will deflect, making the final cut more accurate. Now with the 3/4" piece of plywood cut to final shape, glue the uncut piece of 1/4" plywood on as normal. Wait for the glue to dry, use the jigsaw to cut close to the 3/4" piece (which has now become the template) and repeat the steps above with the pattern bit. Your top is complete.

And it's accurately shaped thanks to your template, which you can save for future cabinets (you know there will be future cabinets, right?). You can do the same to your other pieces. The key is taking the time to make sure your templates are accurate and shaped exactly how you want them.
I know this seems like a lot of steps, but the end result will satisfy you and make it all that much more worth it. I'm a heavy advocate of templates and jigs and fixtures. I often spend more time making jigs, fixtures and templates than I do making projects. The end result is worth it IMHO.
You can also make a template for your corners. Here's how I would do it: Use MDF again; it's easiness of shaping and sanding is what makes it such a great material. You only need one corner with the MDF, so again, take the time to make sure it's *exactly* how you want it. Use whatever method you want to arrive at the desired radius (be it paint can, coffee cup, bottle lid, compass, it doesn't matter so long as it's the desired radius

). Cut close to the line with your trusty jigsaw and then sand it to final shape. A rasp works well when shaping MDF btw. Once you're happy with the shape of the curve, take it over to the workpiece. Lay it down on the workpiece and secure it (in this case I'd go with clamps because you have to move it 4 times). Make sure the edges are even and line up. You're only cutting the curve here, so you don't need to move the router much. You use the same router bit you used when doing the top. Repeat with the other 3 corners. Save the template.
Hope this helps some. If you have any questions about the procedure post away.
I definitely think contact cement is strong enough. It's pretty much a matter of preference now though. If you use the TBII, clamp it good and tight. The joint will never fail if clamped correctly.