Rust is a bicycle frame's nemesis, to say the least. If you can see rust on the outside, be prepared for its presence on the inside of tubes also. Rust, if allowed to run its natural course, will eat a tube set clean through. Even before the rust finishes its work, the tubing will become so weak that it will be hazardous to ride the bicycle.
Surface rust is not a major issue and is often found on exposed metal surfaces, where paint chips, scratches or rub marks have occurred. Even surface rust on chrome plated surfaces is an issue that is not all that difficult to deal with. However, if the oxidation has gone beyond the surface stage and begun to manifest itself in the form of "pitting", the oxidation issues becomes serious and eventually costly. Pitting will usually be accompanied with paint or chrome "peel". If you see rust, coupled with peeling of either paint or chrome, the issue is worth concern. Repair might be both difficult and costly to implement.
And, of course, though it is considerably less common, alloy components can experience the horrors of oxidation also. An oxidized alloy component can often be cleaned up very nicely, if the oxidation is only surface related. Surface related alloy oxidation is commonly referred to as "tarnish" and can be easily polished away. However, pitted alloy components presents an entirely different problem. Once the alloy begins to pit, strength will suffer and, since some alloy parts keep your butt and the road apart, you might want to consider badly oxidized alloy components beyond repair and seek replacement units.
Shiny is nice and most vintage road bicycle will clean up quite well. That said, there will be times when the bicycle's finish is in question. What does one do with a cosmetically challenged bicycle?