It depends on who your hosting company is and how they have you setup, but you could ask for a copy of the web server logs and then look through that and see how much traffic your site is receiving. You could also ask your hosting provider what type of resources they are using to host your site to see how your hardware compares to what its currently running on and what you plan to move it to.
One other thing you can do is turn on performance info ("perfdebug") on our site and examine the data at is being produced on each page load. It will give you an idea of what the server is/was doing when you loaded that page. Again, this depends on how your hosting provider has your site setup (does db reside on same server as web server, etc?).