Man, that looks nice!
I think a good place to start is with the valves, as it's just a 20 minute job to check them.
Pull the tank, remove the valve cover, pull the two plastic plugs out of the left side case and roll it over to top dead center (650s have a "T" mark visible in the small plug hole; 600 should have something similar). You can see the 650's rotor markings here
The cam lobes should be pointing away from one another. If they are not, roll it through another revolution until they are pointing away from one another. That is, the exhaust lobes will point forward and the intake lobes will point aft.
Grab your feeler gauges and check the clearances and we'll go from there.
I think, if the valves are OK, that you'd be looking at testing the petcock for flow, throwing away whatever gas is in the tank and replacing with known good fuel, replacing the fuel line, pulling the spark plug and testing for spark at the plug, and giving the carb a good cleaning. Not all at once, but pretty much in that order.
No fear on the carb; there is a set of videos that can walk you through it.
Logic is this: You looked at it, started it, rode it, and bought it. Who knows how long it sat. Restarting an old, laid up bike can jar stuff loose such that it gives problems after a short ride. If the valves are tight it won't start easily no how. If they are good but the petcock is pooched it won't start with no gas. You may not know the vintage of the gas or what may be in the bottom of the tank; drain it, add a pint of good fuel, swish it around real good, drain it, and stick a gallon of good fuel in there. The fuel line may be 30 years old and degraded, delivering gunk to the carb. If there's no spark at the plug (per Paul's comments) ya gotta get that fixed. If all that passes or gets fixed, the carb may be gunked up. You may have to do only some of these things to get it running, or you may have to do all of them. Just know that it's a simple machine and not a lot can go horribly wrong and nothing is going to cost you a lot of money.
By the way, does it have fresh oil and filter in it?