Also, is there any problem when you install the second MC program in terms of replicating the library I've created on the desktop? If I make changes to the library on one computer, how will I be sure that these changes are also reflected to the library on computer 2?
There are some issues. First of all, each machine does need to have it's own independant library (the database, not the actual media files), unless you use the Media Server (which has it's own limitations). If you try to serve the library files (again, the database, not the media files) over a network, you will probably
run into some serious problems.
If you want to share your media library among multiple machines, but don't want to suffer from the Media Server's limitations (mainly that you can only edit tags on the "host" machine and many media types don't work properly), then
check this thread for my solution...
As far as the hardware... There are lots of options. In your case I would probably look into using some sort of NAS (Network Attached Storage). Either
a standalone device, or (what I would do) by installing more drive space in your existing computer and using it as a server (this is really quite easy to do). Then, you could set up a wireless network in your house, and access your media from anywhere you can get onto your WLAN.
For "travelling", I'd just copy some of your "favorites" over to your laptop's hard drive and set MC up with a separate set of View Schemes that only shows those files. Then, when you're away at Grandma's house, you can still jam out to some tunes!
Hard drive options for laptops are very expensive, and still very limited. About the largest ones currently available are
160GB models, and they carry quite a price premium. Larger ones will come out sooner or later (the new perpendicular recording technology is helping), but why spend a bunch to buy 160GB for a laptop when
you can buy 320GB for less than half the price for a desktop machine?