The simplest thing would be to buy new 10/100 switches and replace your current hubs. That leaves your existing system intact exactly as is but with faster equipment. I recommend switches rather than hubs because, even though they are a little more expensive, they probably provide at least some modest speed advantage.
You would get a router if you want to do some reconfiguring of your system. With a router, you would not need an always-on pc to act as your internet server for the rest of your network. Besides conveninience, it might or might not make your internet sharing go faster. You don't say how you are doing your internet sharing now -- are you just using ICS, or what? I found my internet sharing went much better once I switched to a router. But, if you are happy with your internet sharing now, you don't need to switch (even 10 mbps is much quicker than your isp provides you with).
The router isn't just firewall added (in fact it may not have one, although it has other security features) -- it is dhcp server & NAT, stuff that makes it possible to do internet sharing without using an always-on pc as your server.
So, if the speed problem you are referring to is slow file transfers between machines, then simply getting new switches will help a lot. If the speed problem is you think your internet is too slow, you may not even notice any difference with new switches. A router might help though, because it may be more efficient than your current software. Then again, you may need to get a faster ISP.