I do pretty much this sort of thing every day, we do have rather a lot more machinery though...
Glue: We use a PVA wood glue, panels are spread with the glue and the laminate placed on them, then pressed in a heated press at around 150 bar for about five minutes (depends on the press operator). Normally the laminate is very well stuck afterwards. No preparation of the panels beforehand, we use chipboard (raw and hydrofuge) and MDF.
The shape is then cut on a CNC router (my job) and plastic (PVC, ABS) or laminate edging is applied, either on the router (it's a combined router/edgebander) or a on dedicated edgebander.
For doing it by hand (which we do with panels too thick for the edgebander):
You can either cut your shape first, or apply the laminate first, it depends on the tools you have available, either way, make sure to use a sharp blade when cutting the laminate to avoid chipping it. (One steel blade lasts me about 15-20 panels, average 5 metres of cutting per panel).
I'm assuming you're cutting your shape first.
Cut the shape of your choosing, aim to keep the surfaces to be laminated as smooth and flat as possible, don't polish the edges though, as you need something for the glue to adhere to.
Spread your choice of glue (we use PVA for flat panels, neoprene for edges and curved boards) thinly over the top of the board (start with the bottom first if you're doing both sides). Make sure to remove any impurities in the glue (normally sawdust...). Lay the laminate over the panel (cut your laminate at least 5mm larger than the panel). Smooth it from the middle out toward the edges with your hands.
Now press it... if you're lacking a large press, lay another (thick) board on top and apply pressure as evenly as possible, park your car on it or something else heavy, you can't apply too much pressure.
Leave to dry.
Now you need to trim the laminate to the board. A laminate trimmer or router with the right bit is ideal. In a pinch you could cut it roughly to size with a hacksaw or bandsaw and then file it (gently) to shape. If you have a bandsaw it might be worth making your panel oversize and trying to cut it to shape afterwards, very gently and with a new, fine toothed blade.
Cut strips of laminate for the edges. There is a grain to laminate, the strips should have it running lengthways. Make them slightly larger than your piece (height and length). Apply contact adhesive, wait the required length of time (heating the strips makes them easier to bend and helps the glue set). Start and one end and apply the strip, following it up with a roller, we use the twin-handled sort in order to apply as much pressure as possible.
I can't think of any way to trim the edges other than with a laminate trimmer/router. Plastic is easier as you can file it and use a roundover bit to give a radius). If your tool won't give a 45 degree edge, a plane set at zero depth or even a new machine blade (e.g. plater/thicknesser) can be run along the edge in order to clean it up and chamfer it slightly.
If I were trying this, I'd do the following:
Make the top out of thin (19mm) chipboard, leave it oversize. Apply the laminate. Cut out the finished shape using a router and new bit. Screw & glue battens of chip or MDF to the bottom of the panel to give it the required edge thickness, trim these to shape with a router, using the top as a guide. Apply the edges, finish with a laminate trimmer and a sharp blade.
That's pretty much the approach we use for making 75mm thick worktops, only we use 28mm chip and 45mm MDF, then cut the whole lot in one go on the CNC machine.
An 80*30mm cutter going through 75mm of board a 4m/min makes an awful lot of dust, very quickly.
Hope some of that was helpful.