When someone looks at a classic car for a project how can you check for rust during the purchase process? I've heard of horror stories when someone sends a vehicle off to be sand blasted to only find that it is riddled with rust. Do large patches of surface rust indicate deeper issues?
When purchasing any vehicle, new or old, you will have to inspect it for rust damage. Certain vehicles are known to rust in specific areas due to the nature of their design.
Surface rust may just be that, surface rust. Flash rusting does not necessarily mean that the metal is rotten right through.
Here is an example of flash rust. Most of the time, you can hit this with a wire wheel or flap disc and you will still have structural integrity.
Here is an example of a frame rail with severe rust issues. In the event of purchasing a vehicle with significant damage such as below, you would have to replace the rail completely with new metal.
How can you test areas? Bring a magnet and a body hammer. A magnet will let you know how strong the metal is, and if there is any there. It's also a great way to tell if a car has body fill in it.
Use the narrow end of the body hammer to tap areas in question. You will immediately be able to tell the difference between healthy metal and garbage.
To add to Jonathan Musso's answer - Do your homework. For most older cars, rust patterns are well known, so a bit of time spent perusing owner's club forums etc will enable you to get a good list of places to check for rot (for example, for a Series Land Rover, it's the bulkhead and the chassis outriggers). You'll also be able to get a list of other things to check, such as the mechanical state of the car - many cars will have a particular thing where the marque experts will say "this might only sound like a little rattle, but beware, it's really difficult/expensive to fix, walk away unless you know what you're doing"
Once you've looked at a few, you'll also get to know the obvious places, such as rear wheelarches, where mud and spray from the road accumulate and trap moisture against the metal.
Mostly you'll be looking underneath the car, so take something to lie on (an old coat for example), and a torch.