Biggest problem is still rust, as with any classic. Unless you have the skills/tools to do it yourself by far the most expensive thing is the bodywork, even fairly modest repairs can run to £5k...far more then what (for example) an engine could ever cost you. Stag's are no better or worse then other cars of the time, and generally you can see what rust there is or is not. Also look for all the usual things, good/even panel gaps, tyres wearing evenly to show that the suspension geometry is correct which shows the chassis (well, monocoque) isn't bent etc etc etc. You know all this.
Drive a few. I've driven plenty now and there are plenty of bad ones out there. This includes crap brakes (very common, but easy/cheap to fix), soft/wallowey suspension, over-light steering and dull, flat engines. Like I say, drive some so you know what they are supposed to be like but to be honest they are pretty simple so most things can be sorted fairly easily. Parts supply (except for a few small bits...quarter-light winders, one of the types of water pump) is very good. A good Stag feels quite modern to drive with sharp responses to the controls and confidence inspiring handling.
The engine. Some will tell you they are all fine now, this isn't true. It is only my experience but I have owned two Stag's and arranged to buy a third. Both the Stag's I owned blew their head gaskets a few years after I bought them, the third one blew while it was being prepped. It's not an easy fix because the head bolts are put into the block diagonally so they must come out for the head to be lifted from the block, not easy as aluminium corrosion welds them to the head. It cost's about £2k to replace the head gaskets on a Stag assuming the heads on the car are in good condition. There is no way to tell if a Stag is about to blow it's head gasket, certainly the Triumph specialist I was going to buy the car from wouldn't have bought the one I tried to buy if they knew it was going to go. If I was buying a Stag which had not had it's head gaskets replaced in the last, say, 7 years I would assume it's going to need doing soon.
Upgrades? Never felt the need really. Some people put better radiators in the cars but it's the layout of the cooling system (no header tank, water pump up high) that's the problem, not the capacity....brakes are very good (when in good condition)...yeah I'm not sure what you would change really. I do understand why people but different engines in the car but for me the engine is part of what makes it special so personally I would only have one with the original engine although I concede the Rover is better in every way...
Personally I don't like the auto gearbox, it makes the car a bit soft and flat. The manual O/D box lets you use the engine more, and it is quite a nice engine, far more revvy then you would expect, it feels like a sports car engine. The auto makes the car a soft, comfortable cruiser, the manual makes it more like a sports GT car. It's quite surprising the difference it makes.
I would say the faults are: manual gearbox is a bit slow and agricultural and the well known 'Stag Twitch'. With practice you can drive round the twitch, but the gearbox is never anything other then poor IMHO. The soft top is a bit of a faff to put down (putting it up is fairly easy).