On linux you can hibernate/save the state. Is there someway I can save this to the internet or to a pendrive so that when I use a different PC running linux I can start where I left off?
Short answer: no
Long answer: Well, you could use removable drive of some sort as the swap space to hibernate to, but it would not do you much good. The other computer would have to have the same exact kernel and the same modules, including drivers, loaded. And it would have to be set up to hibernate to the same device, so there would need to be another removable drive on it that you could suspend to and then remove and replace with your drive and then resume from that. And that's just theory, I don't know that it would work even with that.
You might have better luck using Gnome/KDE session restore, but I don't really know what files you might have to shuffle around to do that.
Bottom line: If you really need access to your state, it is probably best to do a remote connection to your computer with ssh, nx, vnc, or rdp.
You can sort of do this kind of thing with some VM software. I believe VMware,virtualbox, qemu/kvm and xen can all do this in one way or another. If you can run your applications inside of a VM this may be an option for you.
Unfortunately I've never seen this kind of thing done on physical machines.