The stetch of the Tweed between Floors and St Boswells is prime fishing territory. Peebles is somewhere that's accessible in a day trip from Edinburgh that you can get to The Tweed easily from. The Ettrick between Selkirk and Galashiels too.
The Tay at Dunkeld was a favourite of my dad's and is reasonably reached from Edinburgh.
To be honest, trout and salmon fishing on these big rivers is a bit of an undertaking if you're travelling with no gear. You'd have to put a bit of work in to rent gear and sort out permits and it takes a bit of experience to fish well casting a fly.
I mentioned trout fisheries as it's a good way for the visitor to have what they need to hand to enjoy a day fishing, including renting tackle. You could try Googling "trout fisheries Scotland" to turn up more if the one I linked to doesn't appeal. Your chances of getting into a fish are much greater if they're stocked lochs.
I'd still recommend a visit to those places on The Tweed I mentioned. It's beautiful country, rarely seen by North American visitors. I've always enjoyed walking and reading the water in big rivers; looking to see what flies are spawning and looking for fish rising to them. I think you can get pleasure from that even if you're not fishing it.
