Hi and welcome!
Sounds like a great find! I hope to someday have a 3A to keep my 4 company (right after I win the lottery!)
I found myself in a similar situation with a motorcycle last summer. It's a 1980 and has all of 7500 miles on it. I started rebuilding the brakes and replacing all rubber lines. It is water cooled, so needed attention there, too, although it wasn't as bad as I feared. I found a lot of congealed gunk in the bottom of the engine, too.
You mentioned some concern about the water jacket of the 3A's engine. It's not likely a big problem. You might just use a garden hose initally, before starting the engine. Run some water through to see what comes out. Just let it run through, don't try to force the full 30-50 psi approx. of most household systems into the radiator, etc. TR3s use a relatively low 4 psi coolant system. Be sure to open the heater valve to let water circulate there, too.
Later, after the engine is started, a more thorough warm flush with a proper system cleaner might be a good idea. Be sure to replace the coolant right away, it's a lubricant for the water pump too, and who knows what's in there now! Incidentally, the most likely rust spot in the TRactor engine is the figure 8 gaskets under the cylinder liners *if* steel gaskets are used. However, most use copper without any problem. If the engine has never been apart, it's most likely copper in there.
If the heater core or the radiator have any build up inside or any small leaks, you'll be happy to know both are pretty easily repaired and rebuildable by most and good radiator shop.
It might be a good idea to check that a thermostat is in place, and, if you wish, test it in a pan of water with a thermometer, to make sure it's working properly. These cars must have a thermostat in place, and hopefully it's the "sleeved" type, that properly covers the bypass once warmed up and open. It sounds like you've already started replacing all the rubber hoses before driving the car much. Good idea!
I agree the brake system should be gone through before the car goes on the road. Be sure to replace all rubber parts. They are no longer trustworthy. If master and slave cylinders are usable, i.e. no pitting in the bores, by all means rebuild instead of sending them off or replacing. In the front calipers, it's possible the pistons are pitted, but unlikely the bores are a problem. I recommend stainless steel replacement pistons.
Yes, grounds should be checked throughout the electrical system. It's often enough to just pull apart the connections and reconnect, to make sure all is working. Any dim bulbs might indicate a socket needs cleaning, too. I hadn't thought about critters chewing on the wiring, as someone else mentioned, but I'm sure anything could happen over the course of 25 years storage!
There are many grease zerks on the car, some in places not found on more modern cars. For example, I think you'll find one on the parkign brake cable, another at brake cable pivot by the rear axle, two outboard on the rear axle just behind the brake drum backing plate and three on the drive shaft. Oh, and I'm not sure about the steering on a TR3, my TR4 has a grease zerk on the rack & pinion.
Others suggest some extra oiling of the engine, since it's been sitting so long. I think that's a good idea too. If it were me, I'd be tempted to pull off the rocker shaft and arms, lift out each pushrod and cam follower one at a time (to avoid any possibiliy of mixing them up) and poor a few spoons full of something like Lucas Oil Treatment or STP right onto the camshaft lobes. The lobes/followers are probably the single most likely areas to wear rapidly and damage in an engine that's been sitting so long. A little, thinner oil in the cylinders to ease the piston rings back into use would be good, too. Then, it might be useful to disonnect the ignition, crank the motor with the starter a little to try to get a little juice flowing inside the engine! And, it would be good to drop the oil pan to clean out any sludge the might have settled, before turning the engine over. Watch the oil pressure closely once the engine is started. It should settle to about 30 psi at idle once warm.
There are "engine flushes" that I have mixed thoughts about. First, they require you run the motor for a short while with the flush replacing some of the oil normally in there. So, essentially you are running a lot of cleaners or detergents and a very lightweight luricant for a short period of time. After a drain, you replace with oil. I am concerned these flushes might move grit and dirt into delicate areas, such as bearing surfaces, that was normally be trapped elsewhere and not be a problem. I also am concerned that some scoring or other problems might result just from using the cleaner, predominantly, as a lubricant for even a short period of time. Also, it's pretty strong stuff and I wonder if it might attack sealer on gaskets from the inside. Finally, the oil that replaces it gets diluted and should also be replaced quickly, I think. Perhaps even two oil changes should be done in quick succession.
You mentioned joining your local Triumph club and that's a great move. Go to the next meeting and ask around for folks who have cars like yours, and see if they would take a look at yours with their experienced eye, and help you spot any concerns. Most will happily do so for the chance to ogle another TR inside and out, or it might cost you a beer and a couple slices of pizza.
As to unleaded gas, don't worry about it. You coiuld likely run the car for 50,000 + miles on unleaded and not notice anything othere than having to adjust the valves a little extra. Eventually, if and when you rebuild the head, have the right kind of valves, valve guides and possibly valve seat inserts put in and it will be ready for another 100,000 miles. I wouldn't bother with lead substitute. You might need to retard the timing a little, to prevent pre-ignition. But, it also depends on the compression ratio of the car. Triumph played around with different compression ratios and even offered compression lowering kits for cars used in countries where gas was poor quality. You might want to try to determine the compression ration on your car... a pressure test on the cylinders would also tell you see if they are all fairly egually balanced. Do further pressure testing, or have it done, if you want to check piston rings, head gasket and the valves.
I did compression tests on the motorcycle I mentioned, and found that despite the low mileage, I probably should rebuild the heads. The valve springs have lost some of their tension, just sitting for all these years. I've not seen this as a common problem on TRs, however.
Heck, you probably want to get out and drive it a little, right? Well, just take care of the safety and critical issues first, then have some fun and fine tune as you go along!
Enjoy!
Alan