OK, if I had just bought a car that had an overheating issue, w/o knowing when that became an issue or anything about the car's history, first thing I'd do is pull the radiator and have it flow checked, and boiled or rodded out if necessary, and if it was bad enough to need more, then a new core or a new radiator. Once you know the radiator is good, you can go from there.
By the way, I have found out that if you remove a radiator and set it aside for a few months without first rinsing it out with fresh water, the damn thing will clog up just sitting there and cause overheating when you put it back in. Happened to me twice. The problem wouldn't show up until I'd been driving for an hour or so. In the first case (slant six), I had the original two row core replaced with a three row and that solved the problem. In the second case (big block that already had a three row core), the radiator was pretty new and had been working fine until I removed it. Having it boiled out solved the problem.
Knock on wood, I suppose, but I've never had a cooling problem that was caused by the fan I was using, or a too-small water pump, or the wrong size pulley or the lack of a shroud. It's always been the radiator - either clogged or just too small for the motor.
