Ooh, sounds like a fun project... one I may be doing myself soon enough.
You will not be able to re-laminate it if it has a bull nose edge, or anything fancy like that unless you plan on cutting that off to make a straight edge.
Not knowing the layout you have, I will imagine it has a lazy Suzan cabinet in some corner. Mine does and I hate it
To recover it with a new laminate, you will need to sand the old laminate top. Nothing too mine, 80/60 grit should work. Sand all the Formica, a random orbit sander would work well, but you will still need to hand-sand some of it I'm sure.
Once it is all sanded, you need to remove any dust./other residue.
Next would be to rough cut your pieces of new laminate.
Once that is done, spread your contact cement (not the little bottle of glue found in school supplies
), lay down some dowels so that when you lay your new Formica up top, you do not contact the glue-prepped counter. Once it's down, it’s pretty much down so being very careful on this. Once you have things lined up properly, start pulling out the dowels, apply pressure by hand and use a roller to get out any remaining air bubbles. Depending on what edge type you want, if doing Formica, start with the front edge first. You will then need a router with a flush cut bit (straight bit/guide roller) and trim the Formica flush with the bottom and top. Then do your top, when you flush cut it, the top sheet will overlap the front edge, helping reduce that black edge on the counter from use. Then doing the back splash use the same principle. Finally cut out your openings for the sink etc with the router(having predrilled a start point first).Another option on the edge trim is to use a piece of wood edging rather than Formica.
While IMHO this isn't a terribly hard project to do, it is time consuming, and does take some elbow grease. Once done though it should give you years of service.
An option to consider would be replacing the countertop in its entirety. If you have a lazy suzan cabinet, a deep counter, odd angles you might just stick with redoing the Formica laminate. But, if you have a simple "L" layout, or straight counters even, the countertop form Home Depot/Lowe’s isn't that terrible, and I think it might be less work overall.
Hope that all made sense, if not I will try to clarify that which does not