Well, since I do web development in ASP.Net, there are a few areas I could see multicore playing a role:
1) Client-side. How can something like Javascript be optimized for the client that has a quad-core CPU if that is what someone wants to harness in running something like sorting a long list of data. Are fat-clients coming back with the new versions of IE, Firefox, Safari and Chrome?
2) Server-side on a web server. Within IIS and the .Net framework that it uses, how do things like PLINQ help use parallel or concurrent programming to help speed up handling requests? What kinds of IIS settings can be done to enhance performance and tune it to the hardware?
3) Middleware/DB Back-end. How does the latest MS-SQL Server or Oracle or MySQL handle using the additional resources of both multi-core and multi-socket, e.g. if a quad-socket motherboard has quad core CPUs in each socket and something like Hyperthreading on top there are 32 threads that could run at once which is really different than a single core CPU back in the days.
In addition, there is something to be said for the multicore aspects of GPUs where Crossfire and SLI were the beginning but now there are more hybrid graphics solutions that one can wonder how this will be harnessed in the future, e.g. AMD's Fusion is one idea that I'm not sure how well it'll do but it is coming last I heard.
On the subject of educating myself, I'm not sure how hard would optimizing my code would help in some cases. I'm more interested in how will IIS try to harness the new computing realm before it as that could ultimately be limiting some things that can be done, even if I isolate my code to run in its own little world.
These are just my current thoughts and are subject to change at any moment.